#6627130

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1315 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)(18,22)(20,28)(23,29)(24,30)(25,32)(27,31);(2,6)(7,8)(9,10,14,12)(11,16,13,15)(18,23)(22,29)(24,25)(30,32);(1,3,5,4)(2,7,6,8)(9,11,14,13)(10,15,12,16)(17,18,20,22,21,23,28,29)(19,25,27,32,26,24,31,30);(1,4,5,3)(2,8,6,7)(9,12,14,10)(11,15,13,16)(17,19,21,26)(18,24,23,25)(20,27,28,31)(22,30,29,32);(9,13,14,11)(10,16,12,15)(19,26)(24,25)(27,31)(30,32)

Polycyclic presentation:

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

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, 6534326);