Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420055 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2476 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15274 |
| prime radical | #60469 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ 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:
(1,2,8,14,7,13,6,12)(3,15,20,9,19,35,18,34)(4,21,26,38,25,10,24,39)(5,27,29,11,31,41,30,42)(16,43,49,59,45,36,22,32)(17,46,28,57,48,33,47,40)(23,52,54,37,55,62,56,50)(44,61,51,63,53,58,64,60);(1,3)(2,9)(4,22)(5,28)(6,20)(7,19)(8,18)(10,36)(11,40)(12,35)(13,34)(14,15)(16,24)(17,29)(21,43)(23,53)(25,49)(26,45)(27,46)(30,48)(31,47)(32,38)(33,41)(37,61)(39,59)(42,57)(44,55)(50,58)(51,54)(52,63)(56,64)(60,62);(1,4)(2,10)(3,16)(5,23)(6,24)(7,25)(8,26)(9,32)(11,37)(12,38)(13,21)(14,39)(15,36)(17,44)(18,22)(19,45)(20,49)(27,52)(28,51)(29,54)(30,56)(31,55)(33,58)(34,59)(35,43)(40,60)(41,62)(42,50)(46,61)(47,64)(48,53)(57,63);(1,5,6,29,7,31,8,30)(2,11,12,41,13,42,14,27)(3,17,18,28,19,48,20,47)(4,23,24,54,25,55,26,56)(9,33,15,40,34,46,35,57)(10,37,38,62,21,50,39,52)(16,44,22,51,45,53,49,64)(32,58,36,60,59,61,43,63)Polycyclic presentation:
2^9|a^2=ef,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,g^b=gi,g^c=gi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327251);