#416177

order	512
nilpotency class	3
isoclinism	isoclinic class of #395552
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1334 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)(4,5)(6,8)(9,10,13,20,15,18,14,19)(11,21,23,16,24,33,30,32)(12,25,28,35,29,36,27,17)(22,37,26,40,39,31,38,34);(2,5)(6,7)(9,11,14,30,15,24,13,23)(10,16,20,33,18,32,19,21)(12,26,28,39,29,38,27,22)(17,34,36,31,35,40,25,37);(9,12)(10,17)(11,22)(13,27)(14,28)(15,29)(16,31)(18,35)(19,25)(20,36)(21,34)(23,38)(24,39)(26,30)(32,37)(33,40);(1,3,4,8)(2,6,5,7)(10,18)(16,32)(17,35)(19,20)(21,33)(25,36)(31,37)(34,40)

Polycyclic presentation:

2^9|a^2=f,b^2=eh,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,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, 323373);