Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24468 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | #34633 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1801 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,14,11,15,9,25)(7,12,18,27,19,29,17,28)(8,20,24,31,22,35,23,13)(16,32,34,36,33,30,21,26);(2,4)(5,7,10,18,11,19,9,17)(6,12,14,27,15,29,25,28)(8,21,24,16,22,34,23,33)(13,30,20,26,31,32,35,36);(5,8)(6,13,15,31)(7,16,19,33)(9,23)(10,24)(11,22)(12,26)(14,20,25,35)(17,21,18,34)(27,32)(28,30)(29,36);(2,4)(8,22)(13,31)(16,33)(20,35)(21,34)(23,24)(26,36)(30,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fg,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72286356, #72286950, #72287097 |
| maximal normal subgroup of | #72286356, #72286950, #72287097 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275973);