#6373852

order	512
nilpotency class	3
isoclinism	isoclinic class of #60457
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	C₂⁴	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1792 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)(3,8)(4,6)(5,7)(9,10,15,20,16,21,14,32)(11,17,26,36,24,35,25,37)(12,27,23,34,29,40,28,18)(13,19,31,38,30,33,22,39);(1,3,5,4)(2,6,7,8)(9,11)(10,17)(12,22)(13,23)(14,25)(15,26)(16,24)(18,33)(19,34)(20,36)(21,35)(27,39)(28,30)(29,31)(32,37)(38,40);(9,12,14,28,16,29,15,23)(10,18,20,27,21,34,32,40)(11,22,25,30,24,31,26,13)(17,33,36,39,35,19,37,38);(1,4,5,3)(2,6,7,8)(9,13,15,31,16,30,14,22)(10,19,20,38,21,33,32,39)(11,23,26,29,24,28,25,12)(17,34,36,40,35,18,37,27);(11,24)(13,30)(17,35)(19,33)(22,31)(25,26)(36,37)(38,39)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6281048);