#6914358

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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1544 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=f,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=efh,d^c=dfh,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, 6821554);