#6474961

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470867
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	#37498	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1440 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)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,27)(20,21)(22,26)(23,25)(24,30)(28,32)(29,31);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,31)(21,32)(22,30)(23,29)(25,28)(26,27);(1,3,5,4)(2,7,6,8)(11,15)(13,16)(17,20,21,18,22,25,23,26)(19,28,32,27,30,31,29,24);(11,15)(13,16)(17,21,22,23)(18,20,26,25)(19,29,30,32)(24,28,27,31);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(17,21,22,23)(18,25,26,20)(19,29,30,32)(24,31,27,28)

Polycyclic presentation:

2^9|c^2=fg,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di

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, 6382157);