#9718232

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

4566 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)(25,26,30,32)(27,31,36,33)(28,34,39,37)(29,38,40,35);(1,3)(2,6)(4,7)(5,8)(11,17)(15,21)(18,23)(22,24)(25,27)(26,31)(28,38)(29,37)(30,36)(32,33)(34,40)(35,39);(10,14)(15,21)(16,19)(22,24)(26,32)(27,36)(34,37)(35,38);(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,27)(26,33)(28,35)(29,37)(30,36)(31,32)(34,40)(38,39);(1,4,5,2)(3,7,8,6)(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,28,30,39)(26,34,32,37)(27,35,36,38)(29,33,40,31);(1,3)(2,7)(4,6)(5,8)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29,30,40)(26,35,32,38)(27,37,36,34)(28,31,39,33)

Polycyclic presentation:

2^9|a^2=gi,e^2=gh,f^2=i,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, 9625428);