#6343637

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343485
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62811	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

4294 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,e^2=i,f^2=i,b^a=bf,e^a=ehi,f^a=fh,f^b=fi,g^b=ghi,d^c=di

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250833);