Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #403928 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2615 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15123 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1797 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)(4,21)(5,25)(6,14)(7,12)(8,13)(9,18)(10,23)(11,27)(16,34)(17,46)(19,33)(20,32)(22,52)(24,39)(26,38)(28,37)(29,36)(30,45)(31,49)(35,54)(40,44)(41,47)(42,61)(43,50)(48,57)(51,53)(55,59)(56,62)(58,63)(60,64);(1,3)(2,9)(4,16)(5,17)(6,19)(7,18)(8,20)(10,34)(11,31)(12,32)(13,33)(14,15)(21,50)(22,42)(23,44)(24,45)(25,41)(26,48)(27,47)(28,49)(29,43)(30,36)(35,58)(37,57)(38,46)(39,40)(51,61)(52,56)(53,63)(54,64)(55,62)(59,60);(1,4)(2,10)(3,16)(5,22)(6,29)(7,23)(8,24)(9,30)(11,35)(12,39)(13,36)(14,21)(15,40)(17,42)(18,44)(19,43)(20,45)(25,51)(26,55)(27,53)(28,54)(31,56)(32,50)(33,34)(37,59)(38,52)(41,60)(46,58)(47,63)(48,62)(49,64)(57,61);(1,5,7,27,8,28,6,26)(2,11,14,25,13,38,12,37)(3,17,18,47,20,49,19,48)(4,22,23,53,24,54,29,55)(9,31,15,41,33,46,32,57)(10,35,21,51,36,52,39,59)(16,42,44,63,45,64,43,62)(30,56,40,60,34,58,50,61)Polycyclic presentation:
2^9|d^2=h,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 318892);