#7080756

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#23858	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1701 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)(11,13)(12,14)(15,16)(17,18,22,26,23,27,21,31)(19,20,29,32,30,28,25,24);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,31)(21,29)(22,25)(23,30)(26,28)(27,32);(2,6)(5,7)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,26)(21,24)(22,32)(23,28)(27,29)(30,31);(3,8)(5,7)(11,16)(13,15)(17,18,21,31,23,27,22,26)(19,28,25,32,30,20,29,24);(10,14)(13,15)(17,21,23,22)(18,26,27,31)(19,25,30,29)(20,32,28,24)

Polycyclic presentation:

2^9|a^2=fgi,c^2=hi,d^2=fg,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,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, 6987952);