#6752739

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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 ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2116 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,26)(19,25,22,23)(20,28,30,24)(27,31,29,32);(1,3)(2,5)(4,8)(6,7)(10,13)(11,15)(17,19,21,22)(18,23,26,25)(20,29,30,27)(24,32,28,31);(9,11,12,15)(10,14,13,16)(18,23)(19,22)(24,31)(25,26)(27,29)(28,32);(2,6)(5,7)(10,13)(14,16)(17,20,21,30)(18,24,26,28)(19,27,22,29)(23,31,25,32);(3,8)(5,7)(17,19,21,22)(18,25,26,23)(20,29,30,27)(24,31,28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6659935);