#382540

order	512
nilpotency class	3
isoclinism	isoclinic class of #188766
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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×#380
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1922 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)(9,10,13,17,14,18,27,34)(11,15,20,29,21,30,38,40)(12,22,24,32,25,33,26,16)(19,35,36,31,23,39,37,28);(2,5)(6,8)(9,11)(10,15)(12,23)(13,20)(14,21)(16,31)(17,29)(18,30)(19,25)(22,39)(24,37)(26,36)(27,38)(28,32)(33,35)(34,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,16)(11,19,21,23)(13,26)(14,25)(15,28,30,31)(17,33)(18,32)(20,37,38,36)(22,34)(24,27)(29,39,40,35);(3,7)(6,8)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289736);