Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219491 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2328 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23426 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1573 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,10,15,11,16,9,13)(7,17,18,26,21,28,20,12)(8,22,24,30,25,29,23,14)(19,33,32,36,35,27,34,31);(1,2,4,3)(5,7,9,20,11,21,10,18)(6,12,15,17,16,26,13,28)(8,19,23,34,25,35,24,32)(14,27,22,31,30,33,29,36);(6,13)(7,18)(9,10)(12,26)(14,29)(15,16)(19,32)(20,21)(22,30)(23,24)(27,33)(34,35);(1,3,4,2)(5,8)(6,14,16,30)(7,19)(9,23)(10,24)(11,25)(12,27,26,33)(13,29,15,22)(17,31,28,36)(18,32)(20,34)(21,35)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=eh,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72226669, #72226673, #72226677 |
| maximal normal subgroup of | #72226669, #72226673, #72226677 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 283392);