Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30056 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2631 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14267 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
948 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,15,19,41)(4,20,24,35)(5,11)(6,12)(7,13)(8,14)(9,18,32,29)(10,23,36,22)(16,30)(17,43,49,62)(21,51,53,37)(25,34,52,61)(26,38)(27,39)(28,40)(31,48,59,55)(33,46)(42,50)(44,56)(45,57)(47,58)(54,63)(60,64);(1,3)(2,9)(4,16)(5,17)(6,29)(7,18)(8,19)(10,33)(11,31)(12,41)(13,15)(14,32)(20,42)(21,44)(22,50)(23,45)(24,46)(25,54)(26,55)(27,48)(28,49)(30,36)(34,60)(35,57)(37,58)(38,62)(39,43)(40,59)(47,52)(51,63)(53,64)(56,61);(1,4)(2,10)(3,16)(5,25)(6,23)(7,22)(8,24)(9,30)(11,37)(12,20)(13,35)(14,36)(15,42)(17,47)(18,50)(19,46)(21,26)(27,53)(28,52)(29,45)(31,58)(32,33)(34,38)(39,61)(40,51)(41,57)(43,56)(44,48)(49,54)(55,64)(59,63)(60,62);(1,5,7,27,8,28,6,26)(2,11,13,39,14,40,12,38)(3,17,18,48,19,49,29,55)(4,21,23,52,24,53,22,25)(9,31,15,43,32,59,41,62)(10,34,20,51,36,61,35,37)(16,44,45,47,46,64,50,54)(30,56,57,58,33,60,42,63)Polycyclic presentation:
2^9|a^2=h,d^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,e^b=ei,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287622);