Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23995 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2487 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
892 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(7,16,19,20)(8,21)(12,17,26,30)(13,27)(18,31,33,34)(22,24)(25,32,35,36)(28,29);(5,6,10,14,11,15,9,23)(7,17,20,26,19,30,16,12)(8,13,22,29,21,27,24,28)(18,32,34,35,33,36,31,25);(5,7,10,16,11,19,9,20)(6,12,14,30,15,26,23,17)(8,18,22,31,21,33,24,34)(13,25,29,36,27,35,28,32);(1,3,4,2)(5,8)(6,13)(7,18)(9,24)(10,22)(11,21)(12,25)(14,29)(15,27)(16,31)(17,32)(19,33)(20,34)(23,28)(26,35)(30,36)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=eg,d^2=eh,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^b=fi,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 | #72265212 |
| maximal normal subgroup of | #72265212 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 286702);