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 | #1445 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1594 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,14,11,16,9,15)(7,17,19,12,20,28,21,27)(8,13,23,31,25,32,24,30)(18,33,35,36,34,29,22,26);(1,2)(3,4)(5,7)(6,12)(8,22,25,35)(9,21)(10,19)(11,20)(13,29,32,33)(14,28)(15,17)(16,27)(18,24,34,23)(26,30,36,31);(2,4)(5,8)(6,13)(7,18,20,34)(9,24)(10,23)(11,25)(12,26,27,36)(14,31)(15,30)(16,32)(17,29,28,33)(19,35,21,22)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=i,f^2=i,g^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 | #72252103, #72252107, #72252113, … |
| maximal normal subgroup of | #72252103, #72252107, #72252113, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31115);