Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #473461 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16866 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
785 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,28,6,12,7,13)(3,9,16,31,17,32,25,37)(4,18,26,10,20,38,21,35)(5,22,23,39,24,11,27,36)(14,40,42,33,43,58,47,29)(15,30,48,59,45,53,46,57)(19,49,50,34,51,60,52,54)(41,62,64,55,44,61,63,56);(1,3,8,16,6,17,7,25)(2,9,13,37,12,32,28,31)(4,14,21,47,20,43,26,42)(5,15,23,48,24,45,27,46)(10,33,38,58,35,29,18,40)(11,30,39,57,22,53,36,59)(19,41,52,63,51,44,50,64)(34,56,60,62,54,55,49,61);(1,4)(2,10)(3,14)(5,19)(6,20)(7,26)(8,21)(9,29)(11,34)(12,35)(13,38)(15,44)(16,47)(17,43)(18,28)(22,54)(23,52)(24,51)(25,42)(27,50)(30,56)(31,58)(32,33)(36,49)(37,40)(39,60)(41,45)(46,63)(48,64)(53,55)(57,62)(59,61);(1,5,7,27,6,24,8,23)(2,11,13,39,12,22,28,36)(3,15,25,46,17,45,16,48)(4,19,21,52,20,51,26,50)(9,30,37,57,32,53,31,59)(10,34,18,49,35,54,38,60)(14,41,47,63,43,44,42,64)(29,55,58,62,33,56,40,61)Polycyclic presentation:
2^9|a^2=ef,b^2=fgh,d^2=efi,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 380658);