#72287296

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	C₂×#33669	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2520 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:

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

Polycyclic presentation:

2^10|c^2=g,d^2=f,e^2=fg,f^2=h,g^2=i,i^2=j,c^b=ch,d^a=dj,d^b=dh,d^c=dgi,e^a=ej,e^b=eh,e^c=ei,e^d=eghi,f^c=fj,f^e=fj,g^d=gi,g^e=gj,i^d=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287390, #72287479, #72320689
maximal normal subgroup of	#72287390, #72287479, #72320689

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