#258120

order	512
nilpotency class	3
isoclinism	isoclinic class of #246159
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

879 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)(9,10,13,16)(11,17,19,14)(12,15,20,23)(18,21,24,22)(25,26)(27,31)(28,30)(29,32)(33,40)(34,37)(35,38)(36,39);(2,5)(6,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,24)(21,23)(26,30)(27,33)(29,35)(31,39)(34,36)(37,40);(9,12)(10,15)(11,18)(13,20)(14,21)(16,23)(17,22)(19,24)(25,27,28,34)(26,31,30,37)(29,36,35,33)(32,39,38,40);(1,3,4,8)(2,6,5,7)(10,16)(12,20)(14,17)(18,24)(26,30)(31,37)(32,38)(39,40)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=ehi,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=dh,g^c=gi

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, 165316);