#407032

order	512
nilpotency class	3
isoclinism	isoclinic class of #399124
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#397
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1118 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)(9,10,14,19)(11,21,24,36)(12,17,28,35)(13,20,15,18)(16,29,31,23)(22,37,38,32)(25,40,39,30)(26,34,27,33);(1,3,4,7)(2,5,6,8)(9,11,15,29,14,24,13,23)(10,16,20,36,19,31,18,21)(12,25,26,38,28,39,27,22)(17,32,33,40,35,37,34,30);(9,12,15,26,14,28,13,27)(10,17,20,33,19,35,18,34)(11,22,29,25,24,38,23,39)(16,30,36,32,31,40,21,37);(2,6)(5,8)

Polycyclic presentation:

2^9|a^2=i,b^2=efh,c^2=ef,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, 314228);