#406830

order	512
nilpotency class	3
isoclinism	isoclinic class of #393989
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#399
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1054 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)(3,5)(4,6)(7,8)(9,10,13,17,14,18,26,33)(11,15,21,29,20,28,37,40)(12,22,24,31,23,32,25,16)(19,34,36,30,35,39,38,27);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,30)(17,29)(18,28)(22,39)(23,35)(24,36)(25,38)(26,37)(27,31)(32,34)(33,40);(9,12,13,25,14,23,26,24)(10,16,17,32,18,31,33,22)(11,19,21,38,20,35,37,36)(15,27,29,39,28,30,40,34);(2,6)(5,8)(11,20)(15,28)(19,35)(21,37)(27,30)(29,40)(34,39)(36,38)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=di,e^c=ei,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, 314026);