Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #464662 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16508 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23434 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1431 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13,8,14,7,12)(3,15,18,9,20,33,19,34)(4,21,24,38,26,37,25,10)(5,11,29,40,30,41,28,42)(16,44,46,59,47,31,22,35)(17,48,51,61,52,43,50,32)(23,53,54,36,55,39,27,56)(45,62,63,64,57,58,49,60);(1,3,7,19,8,20,6,18)(2,9,13,33,14,34,12,15)(4,22,25,47,26,46,24,16)(5,17,28,50,30,52,29,51)(10,35,21,44,38,59,37,31)(11,32,40,48,41,61,42,43)(23,49,27,57,55,63,54,45)(36,60,39,62,56,64,53,58);(1,4,8,26)(2,10,14,38)(3,16,20,47)(5,27,30,54)(6,25,7,24)(9,31,34,44)(11,39,41,53)(12,21,13,37)(15,35,33,59)(17,49,52,63)(18,22,19,46)(23,29,55,28)(32,60,61,64)(36,42,56,40)(43,62,48,58)(45,51,57,50);(1,5)(2,11)(3,17)(4,23)(6,29)(7,28)(8,30)(9,32)(10,36)(12,42)(13,40)(14,41)(15,43)(16,45)(18,51)(19,50)(20,52)(21,39)(22,49)(24,54)(25,27)(26,55)(31,58)(33,48)(34,61)(35,60)(37,53)(38,56)(44,62)(46,63)(47,57)(59,64)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 371859);