#6571293

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1495 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)(9,10,14,13)(11,16,12,15)(17,18,23,26,24,20,22,19)(21,25,30,29,31,28,32,27);(1,3,5,4)(2,6,8,7)(9,10,14,13)(11,16,12,15)(17,19)(18,22)(20,23)(21,29)(24,26)(25,30)(27,31)(28,32);(1,3,5,4)(2,7,8,6)(9,11,14,12)(10,15,13,16)(17,20)(18,24)(19,22)(21,28)(23,26)(25,31)(27,32)(29,30);(1,3,5,4)(2,7,8,6)(9,12,14,11)(10,15,13,16)(17,21,24,31)(18,25,20,28)(19,27,26,29)(22,32,23,30);(1,4,5,3)(2,6,8,7)(9,13,14,10)(11,16,12,15)(17,22,24,23)(18,26,20,19)(21,30,31,32)(25,27,28,29)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6478489);