#328822

order	512
nilpotency class	3
isoclinism	isoclinic class of #326683
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

890 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)(9,10,13,17)(11,18,14,15)(12,20,21,24)(16,22,23,19)(25,26)(27,30)(28,31)(29,32)(33,38)(34,37)(35,39)(36,40);(1,2)(3,5)(4,6)(7,8)(9,11,13,14)(10,15,17,18)(12,19,21,22)(16,20,23,24)(25,27,29,34)(26,30,32,37)(28,33,35,40)(31,38,39,36);(1,3,5,2)(4,6,8,7)(9,12,13,21)(10,16,17,23)(11,19,14,22)(15,20,18,24)(25,28,29,35)(26,31,32,39)(27,33,34,40)(30,36,37,38);(1,4,5,8)(2,6,3,7)(12,21)(16,23)(19,22)(20,24)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 236018);