#293435

order	512
nilpotency class	3
isoclinism	isoclinic class of #272795
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#447	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

991 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)(11,14)(12,15)(13,16)(17,24)(18,22)(19,21)(20,23)(25,26,28,31)(27,30,35,39)(29,37,36,32)(33,38,34,40);(2,5)(6,7)(9,11)(10,14)(12,19)(13,18)(15,21)(16,22)(17,20)(23,24)(25,27)(26,30)(28,35)(29,33)(31,39)(32,40)(34,36)(37,38);(9,12,13,20)(10,15,16,23)(11,17)(14,21)(18,19)(22,24)(26,31)(27,33,35,34)(30,38,39,40)(32,37);(1,3,4,8)(2,6,5,7)(12,20)(15,23)(17,19)(21,24)

Polycyclic presentation:

2^9|a^2=i,c^2=fh,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=dhi,g^c=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 200631);