Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1805 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,8,10)(7,9,11,12)(13,14,17,20)(15,16,19,18)(21,22,26,27)(23,25,31,28)(24,32,33,29)(30,36,34,35);(5,7,8,11)(6,9,10,12)(15,19)(16,18)(22,27)(24,33)(25,28)(30,34);(5,6)(7,9)(8,10)(11,12)(13,15,17,19)(14,18,20,16)(21,23)(22,28)(24,34)(25,27)(26,31)(29,35)(30,33)(32,36);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,16,17,18)(14,19,20,15)(21,24,26,33)(22,29,27,32)(23,30,31,34)(25,35,28,36);(5,6)(7,9)(8,10)(11,12)(13,14)(15,18)(16,19)(17,20)(21,25,26,28)(22,23,27,31)(24,35,33,36)(29,30,32,34)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,c^2=h,d^2=hi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1488874);