#8754843

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

4234 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=i,e^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, 8662039);