#414920

order	512
nilpotency class	3
isoclinism	isoclinic class of #394747
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#394
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

984 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,d^c=di,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, 322116);