#191113

order	512
nilpotency class	3
isoclinism	isoclinic class of #160892
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1068 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)(11,17)(12,20,21,15)(13,16)(14,19)(18,22,24,23)(25,26)(27,33)(28,30)(29,37)(31,35)(32,36)(34,40)(38,39);(2,5)(6,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(26,30)(31,39)(32,37)(33,40);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,27,28,34)(26,31,30,39)(29,38,36,35)(32,40,37,33);(1,3,4,7)(2,6,5,8)(11,19)(14,17)(18,24)(22,23)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=eh,f^2=i,b^a=be,c^a=cf,f^a=fh,d^b=dhi,f^c=fi,g^c=ghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 98309);