#6374254

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342659
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄²

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	C₂×#3449	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4258 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)(4,6)(5,8)(9,10)(11,16)(12,24,27,17)(13,18)(14,19)(15,20)(21,35,37,29)(22,30)(23,31)(25,33,26,32)(28,34)(36,40,38,39);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,21)(13,22)(14,28)(15,23)(17,29)(18,30)(19,34)(20,31)(24,35)(25,36)(26,38)(27,37)(32,39)(33,40);(3,8)(5,7)(9,12)(10,17)(11,21)(13,25)(14,26)(15,27)(16,29)(18,32)(19,33)(20,24)(22,36)(23,37)(28,38)(30,39)(31,35)(34,40);(2,6)(5,7)(9,13,15,14)(10,18,20,19)(11,22,23,28)(12,25,27,26)(16,30,31,34)(17,32,24,33)(21,36,37,38)(29,39,35,40);(9,14,15,13)(10,19,20,18)(11,22,23,28)(12,26,27,25)(16,30,31,34)(17,33,24,32)(21,36,37,38)(29,39,35,40)

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=i,e^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,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, 6281450);