Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394287 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14716 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60387 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2534 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)(3,14)(4,19)(5,11)(6,28)(7,12)(8,13)(9,17)(10,23)(15,30)(16,41)(18,32)(20,34)(21,50)(22,37)(24,36)(25,39)(26,38)(27,40)(29,33)(31,45)(35,51)(42,61)(43,55)(44,56)(46,58)(47,57)(48,52)(49,64)(53,60)(54,63)(59,62);(1,3,6,17,8,18,7,29)(2,9,12,14,13,33,28,32)(4,20,22,15,24,43,23,44)(5,16,25,45,27,47,26,46)(10,34,36,56,37,55,19,30)(11,31,38,41,40,58,39,57)(21,49,51,42,52,62,53,63)(35,59,50,61,60,64,48,54);(1,4)(2,10)(3,15)(5,21)(6,22)(7,23)(8,24)(9,30)(11,35)(12,36)(13,37)(14,34)(16,42)(17,43)(18,44)(19,28)(20,29)(25,51)(26,53)(27,52)(31,54)(32,55)(33,56)(38,50)(39,48)(40,60)(41,59)(45,62)(46,49)(47,63)(57,64)(58,61);(1,5)(2,11)(3,16)(4,21)(6,25)(7,26)(8,27)(9,31)(10,35)(12,38)(13,40)(14,41)(15,42)(17,45)(18,47)(19,48)(20,49)(22,51)(23,53)(24,52)(28,39)(29,46)(30,54)(32,57)(33,58)(34,59)(36,50)(37,60)(43,62)(44,63)(55,64)(56,61)Polycyclic presentation:
2^9|b^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72286066, #72286176, #72286253, … |
| maximal normal subgroup of | #72286066, #72286176, #72286253, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301483);