#382936

order	512
nilpotency class	3
isoclinism	isoclinic class of #188766
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

724 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)(9,10,13,17,14,18,26,33)(11,15,20,28,21,29,36,40)(12,22,25,38,24,16,30,32)(19,34,23,37,35,27,39,31);(2,5)(6,8)(9,11)(10,15)(12,23,24,39)(13,20)(14,21)(16,31,22,37)(17,28)(18,29)(19,25,35,30)(26,36)(27,32,34,38)(33,40);(3,7)(6,8)(9,12)(10,16,18,22)(11,19,21,35)(13,25)(14,24)(15,27)(17,32,33,38)(20,23,36,39)(26,30)(28,31)(29,34)(37,40);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

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

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, 290132);