#8096985

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

4661 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)(3,5,8,7)(9,10)(11,14)(12,16)(13,17)(15,20)(18,22)(19,23)(21,24)(26,32)(28,33)(34,39)(36,40);(1,2,4,6)(3,7,8,5)(9,11,13,19)(10,14,17,23)(12,18,20,24)(15,21,16,22)(25,26,31,32)(27,33,29,28)(30,34,38,39)(35,40,37,36);(1,2)(3,5)(4,6)(7,8)(9,12)(10,15)(11,18)(13,20)(14,21)(16,17)(19,24)(22,23)(25,27,31,29)(26,33,32,28)(30,35,38,37)(34,40,39,36);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,18)(13,19)(15,21)(16,22)(17,23)(20,24)(25,28)(26,27)(29,32)(30,36)(31,33)(34,35)(37,39)(38,40);(9,12)(10,16)(11,18)(13,20)(14,22)(15,17)(19,24)(21,23)(25,29,31,27)(26,28,32,33)(30,35,38,37)(34,40,39,36);(9,10)(11,14)(12,15)(13,17)(16,20)(18,21)(19,23)(22,24)(25,30)(26,34)(27,35)(28,36)(29,37)(31,38)(32,39)(33,40)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004181);