#7026187

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1604 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6933383);