Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #418 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1786 |
| prime radical | #60453 | C2 |
| 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 | |
878 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 878 conjugacy classes of subgroups — too many to draw. Only its 108 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)(11,23,26,52)(12,29,31,18)(13,34)(14,22)(15,21)(16,20)(17,25,41,59)(19,37)(24,51)(27,47,28,53)(30,62,36,67)(32,45,33,46)(35,43,60,42)(38,49)(39,50)(40,54)(44,64,48,65)(55,68)(56,72)(57,71)(58,61)(63,69)(66,70);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,24)(13,35)(14,28)(15,27)(16,26)(18,40)(19,47)(20,41)(21,43)(22,42)(23,50)(25,38)(29,61)(30,63)(31,55)(32,57)(33,56)(34,52)(36,66)(37,59)(39,60)(44,71)(45,70)(46,69)(48,72)(49,53)(51,67)(54,65)(58,64)(62,68);(9,12)(10,18)(11,24)(13,36)(14,32)(15,33)(16,31)(17,40)(19,48)(20,29)(21,45)(22,46)(23,51)(25,58)(26,55)(27,56)(28,57)(30,39)(34,62)(35,66)(37,65)(38,64)(41,61)(42,69)(43,70)(44,49)(47,72)(50,67)(52,68)(53,71)(54,59)(60,63)Polycyclic presentation:
2^9|a^2=f,d^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,h^a=hi,d^b=dh,e^b=ei,f^b=fghi,g^b=gi,h^b=hi,d^c=di,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36971);