#8093844

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

4922 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=gi,e^2=g,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, 8001040);