Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11115 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #388 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1439 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1474 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)(5,6,10,16,11,14,9,15)(7,17,19,31,20,12,25,28)(8,21,23,30,24,32,26,13)(18,33,35,29,22,36,34,27);(5,7)(6,12)(8,22,24,18)(9,19)(10,25)(11,20)(13,29,30,27)(14,17)(15,28)(16,31)(21,33,32,36)(23,35,26,34);(2,4)(5,8)(6,13,14,30)(7,18)(9,23)(10,26)(11,24)(12,27,17,29)(15,21,16,32)(19,34)(20,22)(25,35)(28,36,31,33)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,f^b=fi,g^b=gi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30958);