#72287136

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

2440 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)(6,12)(10,15)(13,16)(17,18,22,23)(19,27,28,24)(20,25,35,36)(21,26,34,37)(29,38,46,41)(30,44,47,39)(31,40,33,43)(32,45,48,42);(1,2)(3,6)(4,8)(5,7)(9,12)(10,13)(11,14)(15,16)(19,28)(24,27)(29,46)(30,47)(32,48)(38,41)(39,44)(42,45);(3,9)(6,12)(10,15)(13,16)(18,23)(19,29)(20,31)(21,34)(24,38)(25,43)(27,41)(28,46)(30,32)(33,35)(36,40)(39,45)(42,44)(47,48);(1,3,4,10,5,9,11,15)(2,6,8,13,7,12,14,16)(19,30,28,47)(20,33)(21,34)(24,39,27,44)(25,43)(26,37)(29,32,46,48)(31,35)(36,40)(38,45,41,42);(2,7)(6,12)(8,14)(13,16)(17,19,31,47,21,32,35,46,22,28,33,30,34,48,20,29)(18,24,40,44,26,42,36,38,23,27,43,39,37,45,25,41)

Polycyclic presentation:

2^10|a^2=j,d^2=f,e^2=gi,f^2=h,g^2=ij,i^2=j,c^a=cj,d^a=dh,d^c=dh,e^a=ej,e^b=ehj,e^c=eg,e^d=egij,f^e=fj,g^c=gi,g^d=gij,i^c=ij,i^d=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287278, #72287657, #72318672
maximal normal subgroup of	#72287278, #72287657, #72318672

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, 1, F.10);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.7*F.9);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);