#409292

order	512
nilpotency class	3
isoclinism	isoclinic class of #394746
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#400
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

982 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)(9,10,15,18,14,19,13,29)(11,20,23,33,24,32,22,16)(12,25,30,34,28,38,27,17)(21,31,37,35,26,39,36,40);(1,3,4,7)(2,5,6,8)(9,11,13,22,14,24,15,23)(10,16,18,20,19,33,29,32)(12,26,27,37,28,21,30,36)(17,31,25,35,34,39,38,40);(9,12,13,27,14,28,15,30)(10,17,18,25,19,34,29,38)(11,21,22,36,24,26,23,37)(16,31,20,35,33,39,32,40);(2,6)(5,8)

Polycyclic presentation:

2^9|a^2=eg,b^2=eh,c^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=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, 316488);