#6379362

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342728
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	C₂⁴	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#125 ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3010 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)(6,7)(9,10)(11,19)(12,23,25,16)(13,17)(14,18)(15,21)(20,28,34,33)(22,31)(24,36,38,30)(26,29)(27,32)(35,39,37,40);(1,2,4,5)(3,7,8,6)(9,11,14,21)(10,15,18,19)(12,20,25,34)(13,26,27,22)(16,28,23,33)(17,31,32,29)(24,37,38,35)(30,40,36,39);(9,12)(10,16)(11,20)(13,24)(14,25)(15,28)(17,30)(18,23)(19,33)(21,34)(22,35)(26,37)(27,38)(29,39)(31,40)(32,36);(1,3,4,8)(2,6,5,7)(11,21)(15,19)(20,34)(22,26)(28,33)(29,31)(35,37)(39,40);(9,13,14,27)(10,17,18,32)(11,22,21,26)(12,24,25,38)(15,29,19,31)(16,30,23,36)(20,35,34,37)(28,39,33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6286558);