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 ◁ C2×#41 ◁ C2×#585 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
4113 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)(3,13,17,36)(4,9)(5,10)(6,24,26,11)(7,12)(8,16)(14,37,39,58)(15,38,21,49)(18,30)(19,31)(20,47,44,32)(22,51,52,34)(23,35)(25,29)(27,41)(28,42)(33,50)(40,59,45,62)(43,55)(46,64,61,57)(48,53)(54,60)(56,63);(1,3,7,17)(2,8,12,29)(4,14,18,39)(5,21,23,15)(6,25,26,16)(9,27,30,53)(10,33,35,28)(11,36,24,13)(19,45,43,40)(20,48,44,41)(22,42,52,50)(31,56,55,54)(32,58,47,37)(34,38,51,49)(46,60,61,63)(57,59,64,62);(1,4)(2,9)(3,14)(5,19)(6,20)(7,18)(8,27)(10,31)(11,32)(12,30)(13,37)(15,40)(16,41)(17,39)(21,45)(22,46)(23,43)(24,47)(25,48)(26,44)(28,54)(29,53)(33,56)(34,57)(35,55)(36,58)(38,59)(42,60)(49,62)(50,63)(51,64)(52,61);(4,18)(9,30)(14,39)(19,43)(20,44)(27,53)(31,55)(32,47)(37,58)(40,45)(41,48)(46,61)(54,56)(57,64)(59,62)(60,63);(1,5,7,23)(2,10,12,35)(3,15,17,21)(4,19,18,43)(6,22,26,52)(8,28,29,33)(9,31,30,55)(11,34,24,51)(13,38,36,49)(14,40,39,45)(16,42,25,50)(20,46,44,61)(27,54,53,56)(32,57,47,64)(37,59,58,62)(41,60,48,63)Polycyclic presentation:
2^9|a^2=g,b^2=i,e^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 | #72302919, #72303034, #72303048, … |
| maximal normal subgroup of | #72302919, #72303034, #72303048, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529666);