#6393869

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342866
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3554	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3027 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,g^c=gi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6301065);