Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2139 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,8,10)(7,11,12,9)(13,14,16,18)(15,19,20,17)(21,22)(23,28)(24,29)(25,30)(26,27)(31,33)(32,36)(34,35);(2,4)(13,14,16,18)(15,17,20,19)(22,27)(23,29)(30,35)(31,36);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,17)(16,18)(19,20)(21,23,26,29)(22,28,27,24)(25,31,34,36)(30,33,35,32);(5,6)(7,9)(8,10)(11,12)(21,24,26,28)(22,29,27,23)(25,33,34,32)(30,31,35,36);(5,7,8,12)(6,9,10,11)(13,15,16,20)(14,17,18,19)(21,25)(22,30)(23,31)(24,32)(26,34)(27,35)(28,33)(29,36)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=f,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1176131);