#415395

order	512
nilpotency class	3
isoclinism	isoclinic class of #399124
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#397
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

756 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=ef,d^2=hi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,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, 322591);