#382939

order	512
nilpotency class	3
isoclinism	isoclinic class of #188768
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×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#382
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

710 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)(9,10,13,18,15,19,14,28)(11,16,22,32,21,31,29,33)(12,23,26,39,25,17,27,35)(20,36,24,38,37,30,40,34);(2,5)(6,8)(9,11)(10,16)(12,24,25,40)(13,22)(14,29)(15,21)(17,34,23,38)(18,32)(19,31)(20,26,37,27)(28,33)(30,35,36,39);(3,7)(6,8)(9,12,14,27,15,25,13,26)(10,17,18,35,19,23,28,39)(11,20,22,24,21,37,29,40)(16,30,33,38,31,36,32,34);(1,3,4,7)(2,6,5,8)(11,21)(16,31)(20,37)(22,29)(24,40)(30,36)(32,33)(34,38)

Polycyclic presentation:

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

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, 290135);