#9105297

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

4518 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=gh,e^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,e^c=egi,e^d=egi,f^d=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, 9012493);