Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #359650 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13867 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62848 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3701 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)(4,18,21,9)(5,22)(6,25)(7,12)(8,16)(10,23)(11,28)(14,38,42,59)(15,30)(17,36)(19,48,50,54)(20,34,53,47)(24,37)(26,31)(27,35)(29,51,32,41)(33,49,46,52)(39,43)(40,55,62,58)(44,57)(45,56)(60,63,64,61);(1,3)(2,8)(4,14)(5,15)(6,26)(7,17)(9,32)(10,30)(11,36)(12,31)(13,25)(16,28)(18,29)(19,40)(20,51)(21,42)(22,39)(23,45)(24,44)(27,43)(33,58)(34,38)(35,57)(37,56)(41,53)(46,55)(47,59)(48,60)(49,63)(50,62)(52,61)(54,64);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,29)(10,33)(11,34)(12,18)(13,38)(15,40)(16,41)(17,42)(22,54)(23,49)(24,50)(25,47)(26,51)(27,52)(28,53)(30,55)(31,32)(35,46)(36,59)(37,48)(39,60)(43,61)(44,62)(45,63)(56,64)(57,58);(1,5,7,24)(2,10,12,35)(3,15,17,44)(4,19,21,50)(6,27,28,23)(8,30,31,57)(9,33,18,46)(11,37,25,22)(13,39,36,56)(14,40,42,62)(16,45,26,43)(20,52,53,49)(29,55,32,58)(34,48,47,54)(38,60,59,64)(41,63,51,61)Polycyclic presentation:
2^9|a^2=h,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^c=hi,e^d=ei77 conjugacy classes — view the full character table.
| maximal subgroup of | #72306450, #72306456, #72306470, … |
| maximal normal subgroup of | #72306450, #72306456, #72306470, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266847);