#6374018

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×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×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1878 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)(9,10,15,20,16,21,14,31)(11,17,26,34,24,33,25,39)(12,27,13,19,30,36,29,18)(22,37,38,35,28,40,23,32);(2,5)(6,8)(9,11)(10,17)(12,28)(13,23)(14,25)(15,26)(16,24)(18,35)(19,32)(20,34)(21,33)(22,30)(27,40)(29,38)(31,39)(36,37);(9,12,14,29,16,30,15,13)(10,18,20,27,21,19,31,36)(11,22,25,23,24,28,26,38)(17,32,34,37,33,35,39,40);(9,13,15,30,16,29,14,12)(10,19,20,36,21,18,31,27)(11,23,26,22,24,38,25,28)(17,32,34,37,33,35,39,40);(1,3,4,7)(2,6,5,8)(11,24)(17,33)(22,28)(23,38)(25,26)(32,35)(34,39)(37,40)

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=h,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ci,e^b=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, 6281214);