Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394328 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2617 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14913 |
| prime radical | #60453 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1385 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)(3,14)(4,20)(5,25)(6,29)(7,12)(8,13)(9,17)(10,24)(11,27)(15,34)(16,44)(18,33)(19,41)(21,50)(22,49)(23,36)(26,38)(28,55)(30,37)(31,48)(32,47)(35,53)(39,43)(40,46)(42,58)(45,54)(51,64)(52,59)(56,60)(57,63)(61,62);(1,3,7,18,8,19,6,17)(2,9,29,41,13,33,12,14)(4,21,23,43,24,48,30,15)(5,16,28,47,27,46,26,45)(10,34,36,31,20,39,37,50)(11,32,55,44,25,54,38,40)(22,51,52,62,53,63,56,42)(35,58,59,57,49,61,60,64);(1,4,6,23,8,24,7,30)(2,10,12,36,13,20,29,37)(3,15,17,21,19,43,18,48)(5,22,26,52,27,53,28,56)(9,31,14,39,33,50,41,34)(11,35,38,59,25,49,55,60)(16,42,45,51,46,62,47,63)(32,57,40,61,54,64,44,58);(1,5)(2,11)(3,16)(4,22)(6,26)(7,28)(8,27)(9,32)(10,35)(12,38)(13,25)(14,40)(15,42)(17,45)(18,47)(19,46)(20,49)(21,51)(23,52)(24,53)(29,55)(30,56)(31,57)(33,54)(34,58)(36,59)(37,60)(39,61)(41,44)(43,62)(48,63)(50,64)Polycyclic presentation:
2^9|b^2=eh,c^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^c=fi,g^c=gi,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, 312679);