Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
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 | 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)(7,15)(8,12)(9,22)(10,14)(11,17)(13,23)(16,30)(18,29)(19,28)(20,34)(21,27)(24,33)(25,32)(26,35)(31,36);(5,7)(6,11,14,25)(8,19)(9,17)(10,18)(12,24,27,36)(13,15,22,29)(16,21)(20,33)(23,32)(26,28,34,30)(31,35);(5,8)(6,12)(7,16,18,19)(9,20)(10,21)(11,24,25,36)(13,26)(14,27)(15,28,29,30)(17,31,32,33)(22,34)(23,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,h^a=hi,c^b=cf,e^b=ei,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72316813, #72316818, #72316849, … |
| maximal normal subgroup of | #72316813, #72316818, #72316849, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31199);