Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #428067 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15377 |
| 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 | |
1125 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)(3,8)(4,17)(5,22)(6,25)(7,12)(9,21)(10,24)(11,27)(13,36)(14,39)(15,42)(16,31)(18,46)(19,48)(20,35)(23,34)(26,33)(28,38)(29,41)(30,44)(32,53)(37,60)(40,57)(43,56)(45,49)(47,59)(50,51)(52,54)(55,63)(58,61)(62,64);(1,3,7,16)(2,8,12,31)(4,18)(5,14,24,41)(6,15,27,44)(9,28)(10,29,22,39)(11,30,25,42)(13,20)(17,36)(19,37)(21,43)(23,40,54,64)(26,38)(32,58)(33,56)(34,57,52,62)(35,46)(45,60)(47,49)(48,59)(50,61)(51,55)(53,63);(1,4)(2,9)(3,13)(5,19)(6,26)(7,20)(8,28)(10,32)(11,35)(12,33)(14,37)(15,43)(16,18)(17,25)(21,27)(22,51)(23,53)(24,49)(29,55)(30,46)(31,56)(34,48)(36,42)(38,44)(39,58)(40,63)(41,47)(45,52)(50,54)(57,60)(59,62)(61,64);(1,5,6,23,7,24,27,54)(2,10,11,34,12,22,25,52)(3,14,15,40,16,41,44,64)(4,19,26,53,20,49,21,50)(8,29,30,57,31,39,42,62)(9,32,35,48,33,51,17,45)(13,37,43,63,18,47,38,61)(28,55,46,60,56,58,36,59)Polycyclic presentation:
2^9|b^2=ei,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=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, 335264);