Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #464757 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16509 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1439 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,13,16,8)(4,9)(5,22)(6,25)(7,11)(10,24)(12,27)(14,28,42,38)(15,29,43,39)(17,30,45,40)(18,47,46,31)(19,32)(20,37)(21,33)(23,36)(26,34)(35,50)(41,54,62,55)(44,61,63,56)(48,59,64,57)(49,60)(51,53)(52,58);(1,3,6,17,7,16,27,45)(2,8,25,40,11,13,12,30)(4,18,20,42,21,46,26,14)(5,15,23,44,24,43,53,63)(9,31,37,28,33,47,34,38)(10,29,51,61,22,39,36,56)(19,48,49,62,50,64,52,41)(32,57,60,54,35,59,58,55);(1,4)(2,9)(3,14)(5,19)(6,26)(7,21)(8,28)(10,35)(11,33)(12,37)(13,38)(15,41)(16,42)(17,46)(18,45)(20,27)(22,32)(23,52)(24,50)(25,34)(29,55)(30,47)(31,40)(36,58)(39,54)(43,62)(44,64)(48,63)(49,53)(51,60)(56,57)(59,61);(1,5,7,24)(2,10)(3,15,16,43)(4,19,21,50)(6,23,27,53)(8,29)(9,32)(11,22)(12,36)(13,39)(14,41,42,62)(17,44,45,63)(18,48,46,64)(20,49,26,52)(25,51)(28,54)(30,56)(31,57)(33,35)(34,58)(37,60)(38,55)(40,61)(47,59)Polycyclic presentation:
2^9|a^2=f,b^2=eg,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,d^c=de,g^c=gi,h^c=hi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72285237, #72286066, #72286090, … |
| maximal normal subgroup of | #72285237, #72286066, #72286090, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 371954);