#9718221

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

4525 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:

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

Polycyclic presentation:

2^9|a^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9625417);