Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1438 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,16)(8,20,23,31)(9,24)(10,14)(11,15)(12,18)(13,22,30,21)(17,29,34,26)(19,27)(25,28)(32,33,36,35);(1,2,3,4)(5,7)(6,12)(8,17)(9,25)(10,18)(11,19)(13,29)(14,27)(15,28)(16,24)(20,36)(21,35)(22,33)(23,34)(26,30)(31,32);(5,8)(6,13)(7,17)(9,22)(10,21)(11,23)(12,26)(14,31)(15,30)(16,32)(18,35)(19,34)(20,24)(25,33)(27,36)(28,29)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,d^b=di,d^c=di,e^c=ei,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72265720, #72265724, #72265727, … |
| maximal normal subgroup of | #72265720, #72265724, #72265727, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30799);