#8695350

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

4658 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8602546);