#6540478

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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	#25775	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1556 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)(4,6)(5,8)(17,18)(19,26)(20,31)(21,27)(22,28)(23,32)(24,29)(25,30);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,20,24,27,23,30)(18,25,32,21,29,31,28,26);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,20,23,19,24,30,22,27)(18,26,28,31,29,21,32,25);(1,4,5,3)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(17,21,24,26)(18,27,29,19)(20,32,30,28)(22,31,23,25);(11,15)(14,16)(17,22,24,23)(18,28,29,32)(19,20,27,30)(21,25,26,31)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6447674);