#259872

order	512
nilpotency class	3
isoclinism	isoclinic class of #245999
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

845 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)(3,7)(4,5)(6,8)(9,10)(11,19)(12,17)(13,16)(14,18)(15,21)(20,24)(22,23)(25,26,29,32)(27,30,34,37)(28,31,35,39)(33,38,40,36);(2,5)(6,7)(9,11,12,21)(10,15,17,19)(13,22,14,20)(16,24,18,23)(25,27)(26,30)(28,33)(29,34)(31,38)(32,37)(35,40)(36,39);(10,16)(11,20)(13,14)(17,18)(21,22)(23,24)(25,28,29,35)(26,31,32,39)(27,33,34,40)(30,36,37,38);(1,3,4,8)(2,6,5,7)(10,17)(15,19)(16,18)(23,24)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,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,f^c=fh,g^c=gh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 167068);