#304483

order	512
nilpotency class	3
isoclinism	isoclinic class of #273051
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

846 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:

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

Polycyclic presentation:

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

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, 211679);