#6410363

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1221 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=g,d^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dfi,e^c=ei,e^d=ei

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, 6317559);