#6403330

order	512
nilpotency class	3
isoclinism	isoclinic class of #62253
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	#61251	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1692 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6310526);