Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9103 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1846 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)(5,6,10,16,11,15,9,14)(7,12,19,27,20,28,18,26)(8,21,22,13,24,30,23,31)(17,32,33,29,34,36,35,25);(1,2,4,3)(5,7)(6,12)(8,17)(9,18)(10,19)(11,20)(13,29)(14,26)(15,28)(16,27)(21,32)(22,33)(23,35)(24,34)(25,31)(30,36);(1,3,4,2)(5,8)(6,13)(7,17)(9,22)(10,23)(11,24)(12,25)(14,30)(15,31)(16,21)(18,33)(19,35)(20,34)(26,32)(27,36)(28,29);(5,9,11,10)(6,14,15,16)(7,18,20,19)(8,22,24,23)(12,26,28,27)(13,30,31,21)(17,33,34,35)(25,32,29,36)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=e,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,f^b=fi,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72313066, #72313132, #72313170 |
| maximal normal subgroup of | #72313066, #72313132, #72313170 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265215);