#6822360

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1517 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,6)(4,5)(7,8)(10,14)(11,12)(18,22)(23,29)(24,26)(30,32);(9,10)(11,15)(12,16)(13,14)(18,23)(19,25)(21,28)(22,29)(24,26)(27,31);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,18,21,23,20,22,28,29)(19,26,25,32,27,24,31,30);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(19,27)(24,26)(25,31)(30,32);(10,14)(15,16)(17,19,28,31,20,27,21,25)(18,24,29,32,22,26,23,30)

Polycyclic presentation:

2^9|c^2=fh,d^2=gi,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313233, #72313579, #72313767
maximal normal subgroup of	#72313233, #72313579, #72313767

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6729556);