#9889462

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

4567 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)(11,15)(12,19)(13,14)(16,21)(17,24)(18,22)(20,23)(25,26,30,32)(27,34,29,31)(28,33,36,39)(35,40,37,38);(1,2,5,7)(3,8,4,6)(9,10)(11,15)(12,16)(13,14)(17,23)(18,22)(19,21)(20,24)(25,27,30,29)(26,31,32,34)(28,35,36,37)(33,38,39,40);(1,3,5,4)(2,6,7,8)(10,14)(15,22)(16,19)(23,24)(26,32)(31,34)(33,39)(38,40);(9,10)(11,15)(12,16)(13,14)(17,23)(18,22)(19,21)(20,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,38)(36,39)(37,40);(1,2)(3,6)(4,8)(5,7)(9,11,13,18)(10,15,14,22)(12,20,21,17)(16,24,19,23)(25,28)(26,33)(27,35)(29,37)(30,36)(31,38)(32,39)(34,40);(1,4,5,3)(2,6,7,8)(9,12)(10,16)(11,17)(13,21)(14,19)(15,23)(18,20)(22,24)(25,29,30,27)(26,34,32,31)(28,35,36,37)(33,38,39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=ghi,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=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, 9796658);