Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153201 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13348 |
| commutator, Frattini, cosocle | C2×#124 | C23 |
| prime radical | C2×H(5,𝔽2) | D4 |
| LCS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×#124 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
488 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 488 conjugacy classes of subgroups — too many to draw. Only its 50 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)(4,19,22,9)(5,23)(6,12)(7,11)(8,16)(10,27)(13,29)(15,45,48,42)(17,43)(18,37)(20,54,55,35)(21,36,28,41)(24,31)(25,38,59,56)(26,40)(30,49,62,57)(32,50)(33,58)(34,46,53,64)(39,60)(44,52)(47,63,51,61);(1,3,7,18)(2,8,11,31)(4,15,22,48)(5,24,27,16)(6,17,29,52)(9,34,19,53)(10,37,23,14)(12,32,13,33)(20,46,55,64)(21,51,28,47)(25,57,59,49)(26,58,60,50)(30,41,62,36)(35,42,54,45)(38,61,56,63)(39,44,40,43);(1,4)(2,9)(3,15)(5,25)(6,28)(7,22)(8,30)(10,38)(11,19)(12,36)(13,41)(14,42)(16,49)(17,51)(18,48)(20,26)(21,29)(23,56)(24,57)(27,59)(31,62)(32,34)(33,53)(35,40)(37,45)(39,54)(43,63)(44,61)(46,50)(47,52)(55,60)(58,64)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dhi,e^a=ei,f^a=fgh,g^a=gi,c^b=cf,d^b=dhi,g^b=gh,d^c=dg,g^c=gi,h^c=hi,f^d=fh,g^f=gi32 conjugacy classes — view the full character table.
| maximal subgroup of | #72279336, #72279356, #72279992, … |
| maximal normal subgroup of | #72279336, #72279356, #72279992, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60398);