#6373969

order	512
nilpotency class	3
isoclinism	isoclinic class of #3055
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	#60688	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1944 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)(9,10,14,18,15,19,13,28)(11,20,24,30,22,34,23,16)(12,17,26,31,27,32,25,38)(21,33,37,39,35,40,36,29);(2,5)(6,8);(9,11,13,23,15,22,14,24)(10,16,18,20,19,30,28,34)(12,21,25,36,27,35,26,37)(17,29,31,33,32,39,38,40);(9,12,14,26,15,27,13,25)(10,17,18,31,19,32,28,38)(11,21,24,37,22,35,23,36)(16,29,20,33,30,39,34,40);(1,3,4,7)(2,6,5,8)(11,22)(16,30)(20,34)(21,35)(23,24)(29,39)(33,40)(36,37)

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,e^c=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239527, #72239730, #72239811, …
maximal normal subgroup of	#72239527, #72239730, #72239811, …
maximal quotient of	C₂×#6373969

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6281165);