#6567953

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533087
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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 ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1857 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=f,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,g^c=gi,e^d=eh

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313734, #72313890, #72313969
maximal normal subgroup of	#72313734, #72313890, #72313969

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6475149);