Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #459756 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2761 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16348 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
904 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,28,6,12,7,13)(3,14,29,32,17,9,18,41)(4,19,26,10,22,38,23,35)(5,11,24,36,25,37,27,39)(15,42,47,33,20,50,44,30)(16,31,45,57,46,40,48,58)(21,52,53,60,54,49,55,34)(43,61,62,59,51,64,63,56);(1,3,8,29,6,17,7,18)(2,9,28,41,12,14,13,32)(4,20,26,44,22,15,23,47)(5,16,24,45,25,46,27,48)(10,33,38,50,35,30,19,42)(11,31,36,57,37,40,39,58)(21,51,53,63,54,43,55,62)(34,59,52,64,60,56,49,61);(1,4)(2,10)(3,15)(5,21)(6,22)(7,26)(8,23)(9,30)(11,34)(12,35)(13,38)(14,33)(16,43)(17,20)(18,47)(19,28)(24,55)(25,54)(27,53)(29,44)(31,56)(32,42)(36,49)(37,60)(39,52)(40,59)(41,50)(45,63)(46,51)(48,62)(57,64)(58,61);(1,5,8,24,6,25,7,27)(2,11,28,36,12,37,13,39)(3,16,29,45,17,46,18,48)(4,21,26,53,22,54,23,55)(9,31,41,57,14,40,32,58)(10,34,38,52,35,60,19,49)(15,43,47,62,20,51,44,63)(30,56,42,61,33,59,50,64)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,h^b=hi,d^c=de,f^c=fi,h^c=hi,e^d=ei,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 366952);