#6386394

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342800
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×#121	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2907 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)(9,10,14,18,13,17,26,32)(11,19,22,35,21,15,28,30)(12,23,25,33,24,16,27,31)(20,29,37,39,36,34,38,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,29)(17,19)(18,30)(23,34)(24,36)(25,37)(26,28)(27,38)(31,39)(32,35)(33,40);(9,12,13,24)(10,16,17,23)(11,20,21,36)(14,27,26,25)(15,29,19,34)(18,33,32,31)(22,38,28,37)(30,40,35,39);(2,5)(6,8)(11,21)(15,19)(20,36)(22,28)(29,34)(30,35)(37,38)(39,40);(1,3,4,7)(2,6,5,8)(10,17)(11,21)(16,23)(18,32)(20,36)(22,28)(31,33)(37,38)

Polycyclic presentation:

2^9|a^2=h,c^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,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, 6293590);