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 | #1439 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1550 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,15,11,14,9,24)(7,16,18,27,19,12,25,28)(8,20,23,30,21,34,22,13)(17,31,33,29,32,36,35,26);(1,2,3,4)(5,7)(6,12)(8,17)(9,18)(10,25)(11,19)(13,29)(14,16)(15,27)(20,31)(21,32)(22,33)(23,35)(24,28)(26,30)(34,36);(5,8,9,22,11,21,10,23)(6,13,15,20,14,30,24,34)(7,17,18,33,19,32,25,35)(12,26,27,36,16,29,28,31)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,g^b=gi,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31094);