Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196266 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35149 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2561 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15,11,12,9,25)(7,16,20,26,18,31,19,13)(8,21,23,29,24,14,22,28)(17,30,34,27,32,36,33,35);(1,2)(3,4)(6,12)(9,10)(13,26)(14,21)(19,20)(22,23)(27,35)(33,34);(2,4)(5,7)(6,13,12,26)(8,17)(9,19)(10,20)(11,18)(14,27,21,35)(15,16,25,31)(22,33)(23,34)(24,32)(28,36,29,30);(5,8,11,24)(6,14,12,21)(7,17,18,32)(9,23,10,22)(13,27,26,35)(15,29,25,28)(16,30,31,36)(19,34,20,33)Polycyclic presentation:
2^9|a^2=h,c^2=fh,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,f^b=fi,h^b=hi,e^c=ei,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72286501, #72286845 |
| maximal normal subgroup of | #72286501, #72286845 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281242);