#415901

order	512
nilpotency class	3
isoclinism	isoclinic class of #394283
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1580 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)(3,7)(4,5)(6,8)(9,10,15,20)(11,21,24,35)(12,17,27,34)(13,19,14,18)(16,28,31,23)(22,32,39,37)(25,30,38,40)(26,36,29,33);(2,5)(6,7)(9,11,13,23,15,24,14,28)(10,16,18,21,20,31,19,35)(12,25,26,22,27,38,29,39)(17,32,33,30,34,37,36,40);(9,12)(10,17)(11,22)(13,26)(14,29)(15,27)(16,30)(18,33)(19,36)(20,34)(21,37)(23,38)(24,39)(25,28)(31,40)(32,35);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

2^9|a^2=i,b^2=eh,d^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,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, 323097);