#6389431

order	512
nilpotency class	3
isoclinism	isoclinic class of #62307
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#60685	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1687 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6296627);