#6990108

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907869
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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 ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1609 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)(3,6)(4,8)(5,7)(9,10,12,14)(11,15,16,13)(18,23)(19,26)(20,22)(24,31)(27,32)(28,29);(1,2,6,3)(4,7,5,8)(9,11,12,16)(10,13,14,15)(17,18)(19,27)(20,30)(21,23)(22,25)(24,29)(26,31)(28,32);(1,3,6,2)(4,7,5,8)(11,16)(13,15)(17,19,20,28,21,29,22,26)(18,24,30,31,23,27,25,32);(1,4,6,5)(2,7,3,8)(10,14)(13,15)(17,20,21,22)(18,25,23,30)(19,28,29,26)(24,32,27,31);(1,5,6,4)(2,7,3,8)

Polycyclic presentation:

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

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, 6897304);