Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124838 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1507 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×#97 | C23 |
| socle | C2 | #7626 |
| prime radical | C2×#575 | C4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
473 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 473 conjugacy classes of subgroups — too many to draw. Only its 76 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)(3,13,16,40,18,41,17,34)(4,19,21,9)(5,23,27,10)(6,12)(7,11)(8,15,30,24,31,46,38,48)(14,39,43,63,44,58,47,61)(20,32)(22,50,52,33)(25,35)(26,36)(28,54,57,37)(29,42,59,55,49,45,60,51)(53,62)(56,64);(1,3)(2,8)(4,14)(5,24)(6,16)(7,18)(9,29)(10,34)(11,31)(12,38)(13,37)(15,28)(17,26)(19,49)(20,51)(21,44)(22,47)(23,40)(25,55)(27,48)(30,36)(32,61)(33,59)(35,63)(39,64)(41,54)(42,53)(43,52)(45,56)(46,57)(50,60)(58,62);(1,4)(2,9)(3,14)(5,25)(6,22)(7,21)(8,29)(10,35)(11,19)(12,33)(13,39)(15,45)(16,47)(17,43)(18,44)(20,27)(23,32)(24,55)(26,52)(28,56)(30,60)(31,49)(34,63)(36,50)(37,64)(38,59)(40,61)(41,58)(42,46)(48,51)(53,57)(54,62)Polycyclic presentation:
2^9|a^2=e,d^2=gi,e^2=h,g^2=i,b^a=bd,c^a=cfh,d^a=df,f^a=fi,d^b=dg,e^b=efg,g^b=gi,h^b=hi,d^c=di,e^c=ei,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32034);