#6372157

order	512
nilpotency class	3
isoclinism	isoclinic class of #60457
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

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	C₂²×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2016 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)(9,10,15,20,16,21,14,30)(11,17,23,32,24,18,12,25)(13,19,29,35,27,34,28,39)(22,31,37,40,36,33,26,38);(2,5)(6,8)(9,11,15,23,16,24,14,12)(10,17,20,32,21,18,30,25)(13,22,29,37,27,36,28,26)(19,31,35,40,34,33,39,38);(9,12,14,24,16,23,15,11)(10,18,20,25,21,17,30,32)(13,26,28,36,27,37,29,22)(19,33,35,38,34,31,39,40);(1,3,4,7)(2,6,5,8)(9,13)(10,19)(11,22)(12,26)(14,28)(15,29)(16,27)(17,31)(18,33)(20,35)(21,34)(23,37)(24,36)(25,38)(30,39)(32,40);(13,27)(19,34)(22,36)(26,37)(28,29)(31,33)(35,39)(38,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,e^d=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, 6279353);