#329033

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

768 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:

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

Polycyclic presentation:

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

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, 236229);