#6567624

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533087
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 ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1435 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,7)(4,6)(5,8)(9,10,14,12)(11,17,19,15)(13,16,21,20)(18,23,24,22)(25,26,30,33)(27,34,36,39)(28,32,29,37)(31,38,40,35);(1,3,5,4)(2,6,8,7)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23);(1,3,5,4)(2,7,8,6)(9,12,14,10)(11,15,19,17)(13,20,21,16)(18,22,24,23)(25,27)(26,31)(28,35)(29,38)(30,36)(32,39)(33,40)(34,37);(1,4,5,3)(2,6,8,7)(13,21)(16,20)(18,24)(22,23)(25,28,30,29)(26,32,33,37)(27,35,36,38)(31,39,40,34);(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=f,d^2=fhi,e^2=h,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,g^c=gi,e^d=eh

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6474820);