#382963

order	512
nilpotency class	3
isoclinism	isoclinic class of #188768
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

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	#2283
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

780 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)(3,6)(4,5)(7,8)(9,10)(11,19,23,33)(12,16)(13,17)(14,18)(15,22,20,31)(21,32,36,40)(24,26)(25,29)(27,30)(28,35,34,39)(37,38);(2,5)(6,8)(9,10)(11,20)(12,24)(13,17)(14,18)(15,23)(16,26)(19,31)(21,28)(22,33)(25,37)(27,30)(29,38)(32,35)(34,36)(39,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15,18,20)(12,21)(13,22)(14,23)(16,28,24,34)(17,19,30,33)(25,35)(26,36)(27,31)(29,32,37,40)(38,39);(3,7)(6,8)(9,12,13,25,14,26,27,38)(10,16,17,29,18,24,30,37)(11,21,22,35,23,36,31,39)(15,28,19,32,20,34,33,40)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=f,f^2=i,b^a=be,c^a=cf,h^a=hi,c^b=cg,d^b=di,h^b=hi,d^c=de,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 290159);