Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143951 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1667 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#135 | D4 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
415 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 415 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 68 points:
(1,2)(3,4)(5,6,14,22)(7,23,28,44)(8,29,36,54)(9,37,11,18)(10,19,40,17)(12,21,13,20)(15,26,46,25)(16,39,52,32)(24,43,60,68)(27,55,42,53)(30,65,58,57)(31,51,33,50)(34,49,35,48)(38,56,41,45)(47,61,66,62)(59,64,63,67);(2,4)(5,7)(6,15)(8,30)(9,38)(10,26)(11,41)(12,42)(13,27)(14,28)(16,47)(17,53)(18,44)(19,55)(20,56)(21,45)(22,46)(23,37)(24,33)(25,40)(29,64)(31,60)(32,63)(34,61)(35,62)(36,58)(39,59)(43,49)(48,68)(50,57)(51,65)(52,66)(54,67);(5,8,14,36)(6,16,22,52)(7,24,28,60)(9,39,11,32)(10,33,40,31)(12,35,13,34)(15,43,46,68)(17,54,19,29)(18,49,37,48)(20,51,21,50)(23,57,44,65)(25,61,26,62)(27,63,42,59)(30,38,58,41)(45,64,56,67)(47,53,66,55)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=gi,e^2=hi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=ei,f^b=fg,g^b=gi,d^c=dg,e^c=eh,f^c=fg,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51149);