#392312

order	512
nilpotency class	3
isoclinism	isoclinic class of #188764
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

1298 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)(9,10,15,20,16,21,14,30)(11,22,24,33,25,18,12,17)(13,27,28,35,29,19,31,36)(23,34,26,40,38,37,39,32);(2,5)(6,8)(9,11,14,12,16,25,15,24)(10,17,20,22,21,33,30,18)(13,23,28,26,29,38,31,39)(19,32,35,37,27,40,36,34);(1,3,4,7)(2,6,5,8)(9,12,14,25,16,24,15,11)(10,18,20,17,21,22,30,33)(13,26,28,38,29,39,31,23)(19,34,35,32,27,37,36,40);(9,13)(10,19)(11,23)(12,26)(14,28)(15,31)(16,29)(17,32)(18,34)(20,35)(21,27)(22,37)(24,39)(25,38)(30,36)(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,c^b=ci,h^d=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299508);