#392311

order	512
nilpotency class	3
isoclinism	isoclinic class of #188761
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1320 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 299507);