Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #359617 |
| 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 | C2×#3586 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3651 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,12,6,27)(3,14,19,40,17,9,18,32)(4,10,22,37,23,35,28,36)(5,24,25,38,26,11,29,39)(15,30,20,49,44,41,47,33)(16,31,46,57,45,42,48,56)(21,51,53,60,54,34,52,59)(43,61,63,64,62,55,50,58);(1,3)(2,9)(4,20)(5,16)(6,19)(7,18)(8,17)(10,33)(11,31)(12,14)(13,40)(15,22)(21,50)(23,47)(24,42)(25,48)(26,45)(27,32)(28,44)(29,46)(30,36)(34,58)(35,49)(37,41)(38,57)(39,56)(43,52)(51,64)(53,62)(54,63)(55,59)(60,61);(1,4)(2,10)(3,15)(5,21)(6,28)(7,22)(8,23)(9,30)(11,34)(12,35)(13,37)(14,41)(16,43)(17,44)(18,47)(19,20)(24,51)(25,53)(26,54)(27,36)(29,52)(31,55)(32,49)(33,40)(38,60)(39,59)(42,61)(45,62)(46,63)(48,50)(56,64)(57,58);(1,5,8,26)(2,11,12,24)(3,16,17,45)(4,21,23,54)(6,25,7,29)(9,31,14,42)(10,34,35,51)(13,38,27,39)(15,43,44,62)(18,46,19,48)(20,50,47,63)(22,52,28,53)(30,55,41,61)(32,56,40,57)(33,58,49,64)(36,59,37,60)Polycyclic presentation:
2^9|a^2=h,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72341336, #72341337, #72341363, … |
| maximal normal subgroup of | #72341336, #72341337, #72341363, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266816);