#6403328

order	512
nilpotency class	3
isoclinism	isoclinic class of #62265
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	#61234	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#436 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1680 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)(11,20,23,30,24,34,22,16)(12,17,27,32,25,31,26,38)(21,33,37,39,35,40,36,29);(1,3)(2,5)(4,8)(6,7)(12,25)(17,31)(21,35)(26,27)(29,39)(32,38)(33,40)(36,37);(9,11,13,22,15,24,14,23)(10,16,18,20,19,30,28,34)(12,21,26,36,25,35,27,37)(17,29,32,33,31,39,38,40);(3,8)(5,7)(12,25)(17,31)(21,35)(26,27)(29,39)(32,38)(33,40)(36,37);(2,6)(5,7)(9,12,14,27,15,25,13,26)(10,17,18,32,19,31,28,38)(11,21,23,37,24,35,22,36)(16,29,20,33,30,39,34,40)

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,d^b=df,e^b=ei,e^d=ei

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, 6310524);