#6834604

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806903
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	#39861	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1766 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)(3,7)(4,6)(5,8)(9,10)(11,16)(12,14)(13,15)(17,18,23,27)(19,24,28,31)(20,25,22,30)(21,26,29,32);(1,3)(2,5)(4,8)(6,7)(17,19)(18,24)(20,29)(21,22)(23,28)(25,32)(26,30)(27,31);(9,11,13,12)(10,14,15,16)(17,20,29,19,23,22,21,28)(18,25,32,24,27,30,26,31);(2,6)(5,7)(17,21,23,29)(18,26,27,32)(19,22,28,20)(24,30,31,25);(3,8)(5,7)(9,12,13,11)(10,14,15,16)(17,22,29,28,23,20,21,19)(18,25,32,24,27,30,26,31)

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=i,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,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, 6741800);