#6393868

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×#3561	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3046 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,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, 6301064);