#72287384

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#7623096	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1451 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2441 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(3,9)(6,10)(11,15)(13,16)(17,18)(19,27)(20,25)(21,26)(22,23)(24,31)(28,38)(29,36)(30,39)(32,45)(33,43)(34,42)(35,37)(40,46)(41,48)(44,47);(1,2)(3,10)(4,8)(5,7)(6,9)(11,16)(12,14)(13,15)(17,18)(19,27)(20,25)(21,26)(22,23)(24,31)(28,38)(29,36)(30,39)(32,45)(33,43)(34,42)(35,37)(40,46)(41,48)(44,47);(1,3,4,11,5,9,12,15)(2,6,8,13,7,10,14,16)(18,23)(19,28,31,46)(20,33)(21,35)(24,38,27,40)(25,42)(29,34)(30,32,47,48)(36,43)(39,41,44,45);(2,7)(6,10)(8,14)(13,16)(17,19,29,46,21,32,34,47,22,31,20,28,35,48,33,30)(18,24,36,38,26,41,42,39,23,27,25,40,37,45,43,44);(2,7)(6,10)(8,14)(13,16)(18,23)(19,30)(20,34)(21,35)(24,39)(25,43)(27,44)(28,32)(29,33)(31,47)(36,42)(38,41)(40,45)(46,48)

Polycyclic presentation:

2^10|c^2=f,d^2=gj,f^2=h,g^2=i,i^2=j,c^a=chj,c^b=chj,d^a=dj,d^b=dhj,d^c=dg,e^a=ej,e^b=ehj,e^d=egj,f^d=fj,g^c=gi,g^e=gij,i^c=ij,i^e=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287431, #72287829, #72318707
maximal normal subgroup of	#72287431, #72287829, #72318707

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7*F.10);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 5, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);