Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23882 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2403 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14171 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1629 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,27)(3,14,17,9,19,32,18,33)(4,20,22,10,23,37,28,36)(5,24,25,11,26,39,29,38)(15,34,44,40,45,30,43,49)(16,31,47,55,48,56,46,41)(21,35,52,58,53,59,51,50)(42,61,62,57,64,60,63,54);(1,3)(2,9)(4,15)(5,16)(6,18)(7,17)(8,19)(10,34)(11,31)(12,32)(13,33)(14,27)(20,49)(21,42)(22,44)(23,45)(24,41)(25,47)(26,48)(28,43)(29,46)(30,36)(35,57)(37,40)(38,56)(39,55)(50,61)(51,63)(52,62)(53,64)(54,59)(58,60);(1,4)(2,10)(3,15)(5,21)(6,28)(7,22)(8,23)(9,30)(11,35)(12,37)(13,36)(14,40)(16,42)(17,44)(18,43)(19,45)(20,27)(24,50)(25,52)(26,53)(29,51)(31,54)(32,49)(33,34)(38,59)(39,58)(41,60)(46,63)(47,62)(48,64)(55,61)(56,57);(1,5,7,25,8,26,6,29)(2,11,12,39,13,38,27,24)(3,16,17,47,19,48,18,46)(4,21,22,52,23,53,28,51)(9,31,32,55,33,56,14,41)(10,35,37,58,36,59,20,50)(15,42,44,62,45,64,43,63)(30,54,49,61,34,57,40,60)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=i,f^2=i,h^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,e^c=ei,f^c=fi,e^d=ei,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72285536, #72285688, #72285996, … |
| maximal normal subgroup of | #72285536, #72285688, #72285996, … |
| maximal quotient of | C2×#375827, #72343019 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 283023);