#9727009

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

4343 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,e^2=gh,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, 9634205);