Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144240 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10320 |
| commutator, Frattini, cosocle | C2×#131 | C23 |
| LCS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
574 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 574 conjugacy classes of subgroups — too many to draw. Only its 62 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)(3,14)(4,8)(5,23)(6,27)(7,12)(9,16)(10,19)(11,25)(13,20)(15,39)(17,47)(18,37)(21,51)(22,30)(24,34)(26,38)(28,44)(29,53)(31,49)(32,52)(33,60)(35,56)(36,61)(40,42)(41,43)(45,59)(46,57)(48,54)(50,55)(58,64)(62,63);(2,8)(4,19)(6,20)(9,32)(10,13)(11,29)(12,30)(14,39)(15,42)(16,24)(17,43)(21,48)(22,49)(23,51)(26,50)(27,31)(28,34)(33,57)(35,58)(36,38)(37,59)(40,41)(44,64)(45,63)(47,62)(52,60)(53,61)(54,55);(1,3)(2,9)(4,15)(5,24)(6,17)(7,18)(8,28)(10,33)(11,37)(12,34)(13,35)(14,23)(16,25)(19,42)(20,43)(21,52)(22,45)(26,56)(27,57)(29,59)(30,32)(31,58)(36,62)(38,47)(39,51)(40,54)(41,55)(44,53)(46,50)(48,64)(49,63)(60,61)Polycyclic presentation:
2^9|d^2=gi,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,g^a=gi,c^b=cf,d^b=dg,e^b=eh,g^b=gi,e^c=ei,h^c=hi,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72281308 |
| maximal normal subgroup of | #72281308 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51436);