#8095785

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4493 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=h,d^2=i,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002981);