Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394756 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2436 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14719 |
| prime radical | #60453 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1577 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)(4,20)(5,11)(6,13)(7,12)(8,14)(9,18)(10,23)(16,32)(17,41)(19,38)(21,49)(22,35)(24,34)(25,39)(26,37)(27,31)(28,36)(29,44)(30,45)(33,50)(40,43)(42,54)(46,56)(47,55)(48,51)(52,58)(53,59)(57,63)(60,62)(61,64);(1,3,6,18,8,19,7,27)(2,9,12,15,14,31,13,38)(4,16,22,43,24,44,23,52)(5,17,25,45,26,47,28,46)(10,32,34,58,35,29,20,40)(11,30,36,41,37,56,39,55)(21,42,50,61,51,63,53,62)(33,57,49,64,59,54,48,60);(1,4,6,22,8,24,7,23)(2,10,12,34,14,35,13,20)(3,16,18,43,19,44,27,52)(5,21,25,50,26,51,28,53)(9,29,15,40,31,32,38,58)(11,33,36,49,37,59,39,48)(17,42,45,61,47,63,46,62)(30,54,41,60,56,57,55,64);(1,5)(2,11)(3,17)(4,21)(6,25)(7,28)(8,26)(9,30)(10,33)(12,36)(13,39)(14,37)(15,41)(16,42)(18,45)(19,47)(20,48)(22,50)(23,53)(24,51)(27,46)(29,54)(31,56)(32,57)(34,49)(35,59)(38,55)(40,60)(43,61)(44,63)(52,62)(58,64)Polycyclic presentation:
2^9|b^2=e,c^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301953);