#326756

order	512
nilpotency class	3
isoclinism	isoclinic class of #326735
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×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1059 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)(3,6)(4,5)(7,8)(9,10,13,17)(11,18,20,24)(12,22,14,16)(15,21,23,19)(25,26)(27,30)(28,31)(29,32)(33,38)(34,37)(35,39)(36,40);(2,5)(6,8)(9,10)(11,18)(12,16)(13,17)(14,22)(15,21)(19,23)(20,24)(25,27)(26,30)(28,33)(29,34)(31,38)(32,37)(35,40)(36,39);(1,3)(2,6)(4,7)(5,8)(9,11,13,20)(10,15,17,23)(12,21,14,19)(16,24,22,18)(25,28,29,35)(26,31,32,39)(27,33,34,40)(30,36,37,38);(3,7)(6,8)(9,12,13,14)(10,16,17,22)(11,19,20,21)(15,18,23,24)

Polycyclic presentation:

2^9|a^2=h,c^2=hi,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=de,f^c=fh,g^c=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 233952);