#7624336

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624184
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #532 ◁ #2931 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1270 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,c^b=ci,d^b=df,e^b=ei,g^c=gh,h^c=hi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287478, #72320157, #72320285, …
maximal normal subgroup of	#72287478, #72320157, #72320285, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531532);