Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #503222 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #25771 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1112 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 146 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)(3,15)(4,20)(5,9)(6,28)(7,13)(8,14)(10,24)(11,19)(12,16)(17,44)(18,41)(21,48)(22,53)(23,55)(25,50)(26,32)(27,33)(29,60)(30,39)(31,54)(34,52)(35,63)(36,57)(37,46)(38,43)(40,45)(42,49)(47,51)(56,59)(58,62)(61,64);(1,3,16,27,8,19,6,5)(2,9,28,11,14,33,12,15)(4,21,42,22,25,17,23,51)(7,18,43,58,30,46,29,26)(10,31,59,34,36,61,35,40)(13,32,60,37,39,62,38,41)(20,47,55,44,50,53,49,48)(24,45,63,64,57,52,56,54);(1,4,7,24)(2,10,13,20)(3,17,18,45)(5,22,26,54)(6,23,29,56)(8,25,30,57)(9,31,32,53)(11,34,37,48)(12,35,38,49)(14,36,39,50)(15,40,41,44)(16,42,43,63)(19,21,46,52)(27,51,58,64)(28,59,60,55)(33,61,62,47);(1,5,6,19,8,27,16,3)(2,11,12,9,14,15,28,33)(4,22,23,21,25,51,42,17)(7,26,29,46,30,58,43,18)(10,34,35,31,36,40,59,61)(13,37,38,32,39,41,60,62)(20,48,49,53,50,44,55,47)(24,54,56,52,57,64,63,45)Polycyclic presentation:
2^9|b^2=ei,c^2=f,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,f^b=fh,g^c=gh,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72303766, #72303792, #72312437, … |
| maximal normal subgroup of | #72303766, #72303792, #72312437, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410419);