Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418851 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2679 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15961 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1059 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,13,7,12,6,29)(3,14,17,33,19,9,18,34)(4,20,24,10,23,41,30,37)(5,25,27,40,28,11,26,39)(15,35,21,31,44,58,45,42)(16,38,46,59,47,32,48,60)(22,51,54,56,53,49,55,36)(43,61,50,57,63,64,52,62);(1,3)(2,9)(4,21)(5,16)(6,17)(7,19)(8,18)(10,35)(11,38)(12,14)(13,33)(15,24)(20,31)(22,52)(23,45)(25,32)(26,46)(27,48)(28,47)(29,34)(30,44)(36,61)(37,58)(39,60)(40,59)(41,42)(43,55)(49,57)(50,53)(51,62)(54,63)(56,64);(1,4,6,30,7,23,8,24)(2,10,29,20,12,37,13,41)(3,15,17,21,19,44,18,45)(5,22,26,55,28,53,27,54)(9,31,34,58,14,42,33,35)(11,36,40,49,25,56,39,51)(16,43,46,50,47,63,48,52)(32,57,60,64,38,62,59,61);(1,5)(2,11)(3,16)(4,22)(6,26)(7,28)(8,27)(9,32)(10,36)(12,25)(13,39)(14,38)(15,43)(17,46)(18,48)(19,47)(20,49)(21,50)(23,53)(24,54)(29,40)(30,55)(31,57)(33,59)(34,60)(35,61)(37,56)(41,51)(42,62)(44,63)(45,52)(58,64)Polycyclic presentation:
2^9|a^2=efh,c^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 357444);