#416094

order	512
nilpotency class	3
isoclinism	isoclinic class of #394797
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1142 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)(3,7)(4,8)(5,6)(9,10)(11,21)(12,26)(13,20)(14,19)(15,18)(16,23)(17,28)(22,31)(24,32)(25,33)(27,37)(29,36)(30,35)(34,38)(39,40);(2,6)(7,8)(9,11,14,24,15,25,13,23)(10,16,20,33,18,32,19,21)(12,27,30,39,29,38,28,22)(17,34,36,40,35,37,26,31);(1,3,5,4)(2,7,6,8)(9,12,13,28,15,29,14,30)(10,17,19,26,18,35,20,36)(11,22,23,38,25,39,24,27)(16,31,21,37,32,40,33,34);(1,4,5,3)(2,7,6,8)(10,18)(16,32)(17,35)(19,20)(21,33)(26,36)(31,40)(34,37)

Polycyclic presentation:

2^9|b^2=eh,c^2=e,d^2=hi,e^2=i,f^2=i,g^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,g^b=gi,g^c=gi

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, 323290);