Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | #120 |
| commutator | C23 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | C8×#120 |
| prime radical | #3025 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
945 conjugacy classes of subgroups; only the 264 normal subgroups are shown.
This group has 945 conjugacy classes of subgroups — too many to draw. Only its 264 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 — 4 generators on 64 points:
(1,2,6,12,7,13,25,35,8,5,11,23,26,24,34,48)(3,9,17,29,18,30,43,53,16,15,28,40,42,41,52,60)(4,10,21,32,22,33,47,55,20,19,31,44,46,45,54,61)(14,36,38,56,39,57,59,64,37,27,49,50,58,51,62,63);(1,3)(2,9)(4,14)(5,15)(6,17)(7,18)(8,16)(10,27)(11,28)(12,29)(13,30)(19,36)(20,37)(21,38)(22,39)(23,40)(24,41)(25,43)(26,42)(31,49)(32,50)(33,51)(34,52)(35,53)(44,56)(45,57)(46,58)(47,59)(48,60)(54,62)(55,63)(61,64);(1,4)(2,10)(3,14)(5,19)(6,21)(7,22)(8,20)(9,27)(11,31)(12,32)(13,33)(15,36)(16,37)(17,38)(18,39)(23,44)(24,45)(25,47)(26,46)(28,49)(29,50)(30,51)(34,54)(35,55)(40,56)(41,57)(42,58)(43,59)(48,61)(52,62)(53,63)(60,64);(1,5,6,23,7,24,25,48,8,2,11,12,26,13,34,35)(3,15,17,40,18,41,43,60,16,9,28,29,42,30,52,53)(4,19,21,44,22,45,47,61,20,10,31,32,46,33,54,55)(14,36,38,56,39,57,59,64,37,27,49,50,58,51,62,63)Polycyclic presentation:
2^9|a^2=g,d^2=g,g^2=h,h^2=i,b^a=be,c^a=cf,f^b=fi,e^c=ei128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402873);