#6343591

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343584
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	C₂×#3489	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

4252 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)(11,21,24,34)(12,17)(13,18)(14,20)(15,19)(16,23,30,28)(22,35,38,39)(25,31)(26,32)(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)(20,34)(25,36)(26,38)(27,37)(31,39)(32,40)(33,35);(9,12,15,26)(10,17,19,32)(11,22,24,38)(13,27,14,25)(16,29,30,40)(18,33,20,31)(21,35,34,39)(23,37,28,36);(9,13,15,14)(10,18,19,20)(11,23,24,28)(12,25,26,27)(16,21,30,34)(17,31,32,33)(22,36,38,37)(29,39,40,35);(10,19)(16,30)(17,32)(18,20)(21,34)(29,40)(31,33)(35,39)

Polycyclic presentation:

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

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, 6250787);