#328223

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

960 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)(11,18,14,15)(12,20,21,24)(16,22,23,19)(25,26)(27,33)(28,31)(29,32)(30,35)(34,37)(36,39)(38,40);(2,5)(6,8)(9,11,13,14)(10,15,17,18)(12,19,21,22)(16,20,23,24)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,40)(37,39);(3,7)(6,8)(9,12)(10,16)(11,19)(13,21)(14,22)(15,20)(17,23)(18,24)(25,28)(26,31)(27,34)(29,36)(30,37)(32,39)(33,38)(35,40);(1,3,4,7)(2,6,5,8)(12,21)(16,23)(19,22)(20,24)(28,36)(31,39)(34,40)(37,38)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=ehi,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, 235419);