#6912802

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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 ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1877 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6819998);