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 | C23×C8 | C23 |
| socle | C22 | #1824 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
504 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 504 conjugacy classes of subgroups — too many to draw. Only its 92 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)(3,7,8,5)(4,6)(9,10,14,21,16,19,15,22)(11,23,26,43,28,17,27,42)(12,29,32,49,30,62,33,61)(13,34,37,20,39,50,38,47)(18,36,44,31,46,65,45,63)(24,52,55,71,53,68,56,72)(25,57,59,41,35,66,60,48)(40,58,69,64,51,67,70,54);(9,11)(10,17)(12,24)(13,35)(14,27)(15,26)(16,28)(18,40)(19,23)(20,48)(21,43)(22,42)(25,39)(29,52)(30,53)(31,64)(32,56)(33,55)(34,57)(36,67)(37,59)(38,60)(41,47)(44,70)(45,69)(46,51)(49,72)(50,66)(54,63)(58,65)(61,71)(62,68);(1,3,4,8)(2,5,6,7)(9,12,14,33,16,30,15,32)(10,18,21,45,19,46,22,44)(11,24,27,55,28,53,26,56)(13,36,37,63,39,65,38,31)(17,40,43,69,23,51,42,70)(20,49,50,29,47,61,34,62)(25,58,60,64,35,67,59,54)(41,71,57,68,48,72,66,52)Polycyclic presentation:
2^9|a^2=f,c^2=dg,e^2=hi,f^2=i,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, 42545);