#393295

order	512
nilpotency class	3
isoclinism	isoclinic class of #32941
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

923 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=ef,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,g^a=gi,g^d=gi

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, 300491);