Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#397 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,16,11,14,9,15)(7,12,20,29,18,27,19,28)(8,21,25,31,23,13,24,32)(17,26,34,35,33,36,22,30);(1,2)(3,4)(5,7)(6,12)(8,22,23,34)(9,19)(10,20)(11,18)(13,30,21,35)(14,27)(15,28)(16,29)(17,24,33,25)(26,31,36,32);(2,4)(5,8)(6,13)(7,17)(9,25)(10,24)(11,23)(12,26)(14,21)(15,32)(16,31)(18,33)(19,34)(20,22)(27,36)(28,35)(29,30);(7,18)(12,27)(17,33)(19,20)(22,34)(26,36)(28,29)(30,35)Polycyclic presentation:
2^9|a^2=h,b^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,d^b=di,f^b=fi,e^c=ei,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72252095, #72252107, #72252109, … |
| maximal normal subgroup of | #72252095, #72252107, #72252109, … |
| maximal quotient of | #72267258 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264954);