Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #245204 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1052 conjugacy classes of subgroups; only the 148 normal subgroups are shown.
This group has 1052 conjugacy classes of subgroups — too many to draw. Only its 148 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12)(3,8,17,32)(4,18,19,52)(5,22,25,42)(7,13,27,11)(9,33,35,20)(10,38,41,26)(14,43,45,51)(15,29,47,31)(16,39,50,60)(21,40,54,53)(23,30,49,61)(24,37,55,36)(28,56,58,46)(34,48,57,62)(44,59,64,63);(1,3,6,17)(2,8,12,32)(4,14,19,45)(5,23,25,49)(7,15,27,47)(9,34,35,57)(10,39,41,60)(11,31,13,29)(16,38,50,26)(18,51,52,43)(20,48,33,62)(21,44,54,64)(22,30,42,61)(24,46,55,56)(28,37,58,36)(40,63,53,59);(1,4,7,21)(2,9,13,37)(3,14,15,44)(5,24,26,20)(6,19,27,54)(8,28,29,57)(10,40,22,18)(11,36,12,35)(16,48,23,46)(17,45,47,64)(25,55,38,33)(30,59,39,43)(31,34,32,58)(41,53,42,52)(49,56,50,62)(51,61,63,60);(3,15)(5,25)(8,29)(10,41)(14,44)(16,49)(17,47)(18,52)(20,33)(22,42)(23,50)(24,55)(26,38)(28,57)(30,60)(31,32)(34,58)(39,61)(40,53)(43,63)(45,64)(46,62)(48,56)(51,59)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,d^b=di,f^b=fi,e^c=eh,f^c=fi,f^d=fh44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 152412);