Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622467 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C23×#99 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C2×D4 | C25 |
| prime radical | #62859 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×Q8 ◁ C22×2−1+4 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
4097 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,13,16,36)(4,9)(5,10)(6,24,26,11)(7,12)(8,17)(14,37,40,58)(15,38,43,60)(18,30)(19,46,48,31)(20,32)(21,33)(22,50,52,34)(23,35)(25,29)(27,41)(28,44)(39,61,42,62)(45,59,49,57)(47,54)(51,56)(53,63)(55,64);(1,3)(2,8)(4,14)(5,15)(6,25)(7,16)(9,27)(10,28)(11,36)(12,29)(13,24)(17,26)(18,39)(19,47)(20,40)(21,42)(22,51)(23,43)(30,53)(31,58)(32,54)(33,55)(34,60)(35,56)(37,46)(38,50)(41,48)(44,52)(45,64)(49,63)(57,62)(59,61);(1,4,7,20)(2,9,12,32)(3,14,16,40)(5,21,23,18)(6,19,26,48)(8,27,29,54)(10,33,35,30)(11,31,24,46)(13,37,36,58)(15,42,43,39)(17,41,25,47)(22,49,52,45)(28,55,56,53)(34,59,50,57)(38,62,60,61)(44,64,51,63);(1,5,7,23)(2,10,12,35)(3,15,16,43)(4,18,20,21)(6,22,26,52)(8,28,29,56)(9,30,32,33)(11,34,24,50)(13,38,36,60)(14,39,40,42)(17,44,25,51)(19,45,48,49)(27,53,54,55)(31,57,46,59)(37,61,58,62)(41,63,47,64);(3,16)(8,29)(13,36)(14,40)(15,43)(17,25)(27,54)(28,56)(37,58)(38,60)(39,42)(41,47)(44,51)(53,55)(61,62)(63,64)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=i,f^2=i,b^a=bf,f^a=fh,h^a=hi,e^b=ei,f^b=fi,g^b=ghi,d^c=di,g^f=gi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72302929, #72303034, #72303038, … |
| maximal normal subgroup of | #72302929, #72303034, #72303038, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529664);