Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #363903 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2297 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #21379 |
| prime radical | #3455 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
755 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,5,12,6,10,24,33,7,11,23,34,25,32,42,36)(3,8,16,27,17,28,46,56,15,29,44,55,14,41,43,59)(4,18,20,35,21,48,50,58,19,9,22,31,49,30,51,57)(13,26,39,52,40,53,62,63,38,47,45,54,37,60,61,64);(1,3,5,16,6,17,24,46,7,15,23,44,25,14,42,43)(2,8,12,27,10,28,33,56,11,29,34,55,32,41,36,59)(4,13,22,45,21,40,51,61,19,38,20,39,49,37,50,62)(9,26,35,54,30,53,58,64,18,47,31,52,48,60,57,63);(1,4)(2,9)(3,13)(5,22)(6,21)(7,19)(8,26)(10,30)(11,18)(12,35)(14,37)(15,38)(16,45)(17,40)(20,23)(24,51)(25,49)(27,54)(28,53)(29,47)(31,34)(32,48)(33,58)(36,57)(39,44)(41,60)(42,50)(43,62)(46,61)(52,55)(56,64)(59,63);(2,10,11,32)(3,14,15,17)(5,23)(9,30,18,48)(12,36,34,33)(13,37,38,40)(16,46,44,43)(20,22)(24,42)(27,55)(31,58,35,57)(39,62,45,61)(50,51)(52,54)(56,59)(63,64)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=f,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dhi,f^a=fi,d^b=dh,f^b=fi,e^c=ei,g^c=gi,g^d=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72239242, #72249759, #72249766, … |
| maximal normal subgroup of | #72239242, #72249759, #72249766, … |
| maximal quotient of | #72239244, #72239247, #72239251, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402702);