#8755454

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

4185 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=ghi,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662650);