#8509949

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

4531 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8417145);