Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420056 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15274 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1323 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,30,7,13,6,12)(3,14,29,9,18,41,17,34)(4,19,24,37,23,10,22,38)(5,25,27,54,28,11,31,40)(15,35,20,49,45,42,44,32)(16,46,26,33,47,39,56,58)(21,48,51,61,53,36,52,60)(43,59,55,57,63,62,50,64);(1,3,7,18)(2,9,13,34)(4,20,23,44)(5,26,28,56)(6,29,8,17)(10,35,19,42)(11,39,25,46)(12,41,30,14)(15,22,45,24)(16,31,47,27)(21,43,53,63)(32,37,49,38)(33,54,58,40)(36,57,48,64)(50,52,55,51)(59,61,62,60);(1,4)(2,10)(3,15)(5,21)(6,22)(7,23)(8,24)(9,32)(11,36)(12,37)(13,19)(14,42)(16,43)(17,44)(18,45)(20,29)(25,48)(26,55)(27,51)(28,53)(30,38)(31,52)(33,57)(34,49)(35,41)(39,62)(40,60)(46,59)(47,63)(50,56)(54,61)(58,64);(1,5,6,27,7,28,8,31)(2,11,12,40,13,25,30,54)(3,16,17,26,18,47,29,56)(4,21,22,51,23,53,24,52)(9,33,14,39,34,58,41,46)(10,36,37,60,19,48,38,61)(15,43,44,55,45,63,20,50)(32,57,42,62,49,64,35,59)Polycyclic presentation:
2^9|a^2=ef,b^2=i,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,g^c=gi,h^c=hi,f^d=fi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327253);