Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11000 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #390 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1438 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(5,6)(7,16)(8,13)(9,23)(10,14)(11,15)(12,18)(17,29)(19,28)(20,36)(21,31)(22,32)(24,27)(25,30)(26,34)(33,35);(5,7)(6,12)(8,20,22,34)(9,24)(10,18)(11,19)(13,29,32,33)(14,28)(15,27)(16,23)(17,21,35,25)(26,31,36,30);(5,8)(6,13)(7,17,19,35)(9,25)(10,21)(11,22)(12,26,27,36)(14,31)(15,32)(16,33,28,29)(18,20,24,34)(23,30)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,d^b=di,e^b=ei,d^c=di,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72211251, #72317312, #72317316, … |
| maximal normal subgroup of | #72211251, #72317312, #72317316, … |
| maximal quotient of | #72211196 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31114);