Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #418 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1824 |
| prime radical | C24 | #103 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
343 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 343 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 72 points:
(1,2,4,6,3,5,7,8)(9,10)(11,23,26,17)(12,29,30,61)(13,34)(14,22)(15,21)(16,19)(18,31,44,63)(20,37)(24,52)(25,57,35,66)(27,42,28,43)(32,49,33,62)(36,46,65,45)(38,47)(39,50)(40,54)(41,59,48,60)(51,64)(53,68)(55,72)(56,71)(58,69)(67,70);(2,5)(6,8)(9,11)(10,17)(12,24)(13,35)(14,27)(15,28)(16,26)(18,40)(19,23)(20,48)(21,42)(22,43)(25,39)(29,52)(30,53)(31,64)(32,55)(33,56)(34,57)(36,67)(37,60)(38,59)(41,47)(44,51)(45,69)(46,70)(49,72)(50,66)(54,63)(58,65)(61,68)(62,71);(2,5)(4,7)(9,12,14,33,16,30,15,32)(10,18,22,45,19,44,21,46)(11,24,27,56,26,53,28,55)(13,36,37,63,39,65,38,31)(17,40,43,69,23,51,42,70)(20,49,50,29,47,62,34,61)(25,58,59,64,35,67,60,54)(41,71,57,68,48,72,66,52)Polycyclic presentation:
2^9|a^2=f,c^2=dg,e^2=hi,f^2=gh,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,h^a=hi,e^b=ei,f^b=fi,d^c=di,e^c=eh,f^c=fghi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42621);