#6819027

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	#39189	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1446 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)(11,14)(12,16)(13,15)(17,18,23,27)(19,24,20,30)(21,25,28,32)(22,26,29,31);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,19,23,20)(18,24,27,30)(21,22,28,29)(25,26,32,31);(9,12,13,11)(10,14,15,16)(17,20,21,22,23,19,28,29)(18,24,25,31,27,30,32,26);(2,6)(5,7)(17,21,23,28)(18,25,27,32)(19,22,20,29)(24,26,30,31);(3,8)(5,7)(17,22,21,19,23,29,28,20)(18,26,25,24,27,31,32,30)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=i,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,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, 6726223);