Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133841 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8541 |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | #60402 | C2 |
| LCS | G ▷ C22×Q8 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
1385 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 1385 conjugacy classes of subgroups — too many to draw. Only its 88 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 — 3 generators on 64 points:
(1,2)(3,13,17,35)(4,18,21,38)(5,22,25,10)(6,11)(7,12)(8,15)(9,20,34,27)(14,43,46,63)(16,48,26,42)(19,52)(23,32)(24,39,28,36)(29,40)(30,58)(31,50)(33,54)(37,47)(41,49,61,45)(44,59)(51,53)(55,62)(56,57)(60,64);(1,3)(2,8)(4,14)(5,23)(6,26)(7,17)(9,30)(10,35)(11,37)(12,32)(13,22)(15,25)(16,29)(18,51)(19,53)(20,49)(21,46)(24,47)(27,45)(28,50)(31,40)(33,61)(34,60)(36,42)(38,59)(39,48)(41,57)(43,62)(44,55)(52,63)(54,58)(56,64);(1,4)(2,9)(3,14)(5,19)(6,27)(7,21)(8,30)(10,33)(11,38)(12,34)(13,41)(15,44)(16,49)(17,46)(18,40)(20,29)(22,57)(23,53)(24,56)(25,55)(26,45)(28,54)(31,51)(32,60)(35,61)(36,52)(37,59)(39,62)(42,63)(43,48)(47,64)(50,58)Polycyclic presentation:
2^9|a^2=f,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=di,e^b=ei,f^b=fgi,h^b=hi,e^c=ei,f^c=fhi,g^c=gi,e^d=ei,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41037);