#9127610

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

4498 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9034806);