#6374015

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	C₄²	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#60673	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1898 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)(3,6)(4,5)(7,8)(9,10,15,20,16,21,14,31)(11,17,25,33,26,34,24,39)(12,27,13,19,29,36,28,18)(22,37,30,35,38,40,23,32);(2,5)(6,8)(9,11)(10,17)(12,22)(13,30)(14,24)(15,25)(16,26)(18,32)(19,35)(20,33)(21,34)(23,28)(27,37)(29,38)(31,39)(36,40);(9,12,14,28,16,29,15,13)(10,18,20,27,21,19,31,36)(11,22,24,23,26,38,25,30)(17,32,33,37,34,35,39,40);(9,13,15,29,16,28,14,12)(10,19,20,36,21,18,31,27)(11,23,25,22,26,30,24,38)(17,32,33,37,34,35,39,40);(1,3,4,7)(2,6,5,8)

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,d^b=di

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, 6281211);