#6537225

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ 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)(3,8)(4,6)(5,7)(10,13)(14,16)(19,25)(21,29)(23,31)(24,28)(26,32)(27,30);(1,3,5,4)(2,6,7,8)(9,10,12,13)(11,14,15,16)(17,18,22,20)(19,26,27,23)(21,24,29,28)(25,32,30,31);(10,13)(14,16)(17,19)(18,23)(20,26)(21,30)(22,27)(24,32)(25,29)(28,31);(1,4,5,3)(2,6,7,8)(11,15)(14,16)(17,20,22,18)(19,26,27,23)(21,28,29,24)(25,32,30,31);(9,11,12,15)(10,14,13,16)(18,20)(23,26)(24,28)(31,32)

Polycyclic presentation:

2^9|b^2=fi,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ei,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6444421);