#9781298

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

4654 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)(11,18)(12,20)(13,16)(14,17)(15,22)(19,21)(23,24)(25,26)(27,34)(28,36)(29,32)(30,31)(33,38)(35,39)(37,40);(2,5)(6,7)(9,11,14,16)(10,13,17,18)(12,19,22,23)(15,21,20,24)(25,27,31,29)(26,32,30,34)(28,35,38,37)(33,39,36,40);(1,2)(3,6)(4,5)(7,8)(25,27,31,29)(26,32,30,34)(28,37,38,35)(33,40,36,39);(9,12)(10,15)(11,19)(13,21)(14,22)(16,23)(17,20)(18,24)(25,28,31,38)(26,33,30,36)(27,35,29,37)(32,39,34,40);(1,2)(3,7)(4,5)(6,8)(9,13,14,18)(10,16,17,11)(12,21,22,24)(15,23,20,19)(25,29,31,27)(26,32,30,34)(28,37,38,35)(33,39,36,40);(1,3)(2,6)(4,8)(5,7)(9,10)(11,13)(12,15)(14,17)(16,18)(19,21)(20,22)(23,24)(25,30,31,26)(27,34,29,32)(28,36,38,33)(35,40,37,39)

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=h,e^2=g,f^2=h,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, 9688494);