#415202

order	512
nilpotency class	3
isoclinism	isoclinic class of #394463
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×#397
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

768 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)(11,20)(12,17)(13,28)(14,19)(15,18)(16,22)(21,36)(23,37)(24,31)(25,30)(26,34)(27,35)(29,33)(32,38)(39,40);(9,11,15,24,14,23,13,22)(10,16,28,37,19,31,18,20)(12,25,26,38,27,39,29,21)(17,32,33,30,35,36,34,40);(9,12,13,26,14,27,15,29)(10,17,18,33,19,35,28,34)(11,21,22,25,23,38,24,39)(16,30,20,36,31,40,37,32);(1,3,4,8)(2,5,6,7)

Polycyclic presentation:

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

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, 322398);