#9236998

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

4616 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)(11,14)(12,19)(13,16)(15,20)(17,21)(18,24)(22,23)(26,32)(27,30)(28,34)(37,39);(1,2)(3,7)(4,5)(6,8)(11,17)(14,21)(18,23)(22,24)(25,26)(27,35)(28,29)(30,33)(31,32)(34,40)(36,37)(38,39);(2,5)(6,7)(9,10,13,16)(11,14,17,21)(12,15,20,19)(18,22,23,24)(25,27,31,30)(26,33,32,35)(28,38,34,36)(29,37,40,39);(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(25,28,31,34)(26,29,32,40)(27,36,30,38)(33,39,35,37);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,29)(26,34)(27,37)(28,32)(30,39)(31,40)(33,36)(35,38);(1,3)(2,6)(4,8)(5,7)(9,12)(10,15)(11,18)(13,20)(14,22)(16,19)(17,23)(21,24)(25,30,31,27)(26,33,32,35)(28,38,34,36)(29,39,40,37)

Polycyclic presentation:

2^9|c^2=h,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi

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, 9144194);