#6560576

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	#25392	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1644 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fgi,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ei,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, 6467772);