Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #423176 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15312 |
| 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 | |
1264 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13)(3,15,19,42)(4,10,24,38)(5,25,29,40)(6,14,8,12)(9,30,33,18)(11,27,41,28)(16,34,46,59)(17,32,47,55)(20,31,45,60)(21,50,53,56)(22,37,23,36)(26,43,48,39)(35,52,63,54)(44,62,51,58)(49,61,57,64);(1,3)(2,9)(4,20)(5,26)(6,30)(7,19)(8,18)(10,34)(11,39)(12,42)(13,33)(14,15)(16,23)(17,27)(21,51)(22,46)(24,45)(25,55)(28,47)(29,48)(31,37)(32,40)(35,62)(36,60)(38,59)(41,43)(44,53)(49,52)(50,64)(54,57)(56,61)(58,63);(1,4,8,22,7,24,6,23)(2,10,14,36,13,38,12,37)(3,16,30,45,19,46,18,20)(5,21,27,54,29,53,28,52)(9,31,42,59,33,60,15,34)(11,35,40,50,41,63,25,56)(17,44,26,57,47,51,48,49)(32,58,39,64,55,62,43,61);(1,5)(2,11)(3,17)(4,21)(6,28)(7,29)(8,27)(9,32)(10,35)(12,25)(13,41)(14,40)(15,43)(16,44)(18,48)(19,47)(20,49)(22,54)(23,52)(24,53)(26,30)(31,58)(33,55)(34,61)(36,50)(37,56)(38,63)(39,42)(45,57)(46,51)(59,64)(60,62)Polycyclic presentation:
2^9|a^2=i,c^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330373);