Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #397 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1445 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,10,15,11,14,9,25)(7,16,18,28,19,12,26,29)(8,20,24,30,22,35,23,13)(17,32,33,31,21,34,36,27);(5,7)(6,12)(8,21,22,17)(9,18)(10,26)(11,19)(13,27,30,31)(14,16)(15,28)(20,34,35,32)(23,36,24,33)(25,29);(1,2,3,4)(5,8,9,23,11,22,10,24)(6,13,15,20,14,30,25,35)(7,17,18,33,19,21,26,36)(12,27,28,34,16,31,29,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,f^b=fi,g^b=gi,d^c=di,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31138);