#6537205

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533105
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 ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1544 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)(17,18)(19,27)(20,25)(21,23)(22,26)(24,28)(29,32)(30,31);(1,3,5,4)(2,6,7,8)(9,10,12,13)(11,14,15,16)(18,23)(24,31)(25,26)(27,32);(10,13)(14,16)(17,19,21,29)(18,24,23,31)(20,28,22,30)(25,32,26,27);(1,4,5,3)(2,6,7,8)(11,15)(14,16)(17,20,21,22)(18,25,23,26)(19,28,29,30)(24,32,31,27);(9,11,12,15)(10,14,13,16)(17,20,21,22)(18,26,23,25)(19,28,29,30)(24,27,31,32)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,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, 6444401);