#9267049

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

4460 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9174245);