Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #245105 |
| 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 | |
1060 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 1060 conjugacy classes of subgroups — too many to draw. Only its 154 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:
(1,2,7,12)(3,8,15,28)(4,18,21,34)(5,22,25,10)(6,11,13,26)(9,23,33,20)(14,39,40,56)(16,44,43,29)(17,30,36,53)(19,49,37,62)(24,51,38,35)(27,45,55,42)(31,59,60,64)(32,52,54,50)(41,48,63,61)(46,58,47,57);(1,3,13,36)(2,8,26,53)(4,14,37,63)(5,16,38,46)(6,17,7,15)(9,31,54,55)(10,29,51,57)(11,30,12,28)(18,48,62,56)(19,41,21,40)(20,42,52,59)(22,44,35,58)(23,45,50,64)(24,47,25,43)(27,33,60,32)(34,61,49,39);(1,4,7,21)(2,9,12,33)(3,14,15,40)(5,23,25,20)(6,19,13,37)(8,27,28,55)(10,34,22,18)(11,32,26,54)(16,45,43,42)(17,41,36,63)(24,52,38,50)(29,56,44,39)(30,31,53,60)(35,62,51,49)(46,64,47,59)(48,58,61,57);(3,15)(5,24)(8,28)(10,35)(14,40)(16,46)(17,36)(18,49)(20,50)(22,51)(23,52)(25,38)(27,55)(29,57)(30,53)(31,60)(34,62)(39,61)(41,63)(42,59)(43,47)(44,58)(45,64)(48,56)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gh,c^b=cg,d^b=di,e^c=eh,f^c=fi,f^d=fh50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 152317);