Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394481 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2472 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14737 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1100 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)(3,15)(4,10)(5,26)(6,14)(7,13)(8,12)(9,18)(11,27)(16,34)(17,45)(19,33)(20,32)(21,49)(22,51)(23,37)(24,36)(25,38)(28,40)(29,39)(30,43)(31,47)(35,52)(41,46)(42,61)(44,56)(48,57)(50,64)(53,59)(54,60)(55,63)(58,62);(1,3,8,20,7,19,6,18)(2,9,14,33,13,32,12,15)(4,21,23,16,25,43,24,44)(5,17,29,48,28,47,27,46)(10,34,36,49,38,56,37,30)(11,31,40,57,39,45,26,41)(22,50,52,42,54,62,53,63)(35,58,51,61,59,64,60,55);(1,4,6,23,7,25,8,24)(2,10,12,36,13,38,14,37)(3,16,18,43,19,44,20,21)(5,22,27,52,28,54,29,53)(9,30,15,34,32,49,33,56)(11,35,26,51,39,59,40,60)(17,42,46,62,47,63,48,50)(31,55,41,58,45,61,57,64);(1,5)(2,11)(3,17)(4,22)(6,27)(7,28)(8,29)(9,31)(10,35)(12,26)(13,39)(14,40)(15,41)(16,42)(18,46)(19,47)(20,48)(21,50)(23,52)(24,53)(25,54)(30,55)(32,45)(33,57)(34,58)(36,51)(37,60)(38,59)(43,62)(44,63)(49,61)(56,64)Polycyclic presentation:
2^9|b^2=ef,c^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^c=fi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303919);