#6912727

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
prime radical	#29602	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1715 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)(17,18)(19,20)(21,24)(22,28)(23,25)(26,27)(29,30)(31,32);(2,6)(4,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,22)(18,20,25,28)(21,29,31,26)(24,30,32,27);(1,3)(2,7)(4,6)(5,8)(9,11)(10,13)(12,16)(14,15)(17,20,19,18,23,28,22,25)(21,30,26,32,31,27,29,24);(2,6)(3,8)(11,16)(13,15)(17,21,23,31)(18,24,25,32)(19,26,22,29)(20,27,28,30);(1,4,5,7)(2,3,6,8)(10,14)(13,15)(21,31)(24,32)(26,29)(27,30)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=i,e^2=f,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^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, 6819923);