Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460459 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16364 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
893 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,7,11,27,38)(3,13,16,39,17,8,25,30)(4,9,21,34,20,33,26,40)(5,22,23,52,24,10,35,37)(14,41,18,48,44,28,47,31)(15,29,45,57,46,42,53,62)(19,32,51,59,50,36,54,61)(43,56,49,60,63,55,64,58);(1,3,7,17)(2,8,11,13)(4,18,20,47)(5,15,24,46)(6,25,27,16)(9,31,33,48)(10,29,22,42)(12,39,38,30)(14,26,44,21)(19,49,50,64)(23,53,35,45)(28,40,41,34)(32,58,36,60)(37,62,52,57)(43,54,63,51)(55,61,56,59);(1,4)(2,9,11,33)(3,14)(5,19)(6,26)(7,20)(8,28,13,41)(10,36,22,32)(12,40,38,34)(15,43)(16,47)(17,44)(18,25)(21,27)(23,54)(24,50)(29,56,42,55)(30,48,39,31)(35,51)(37,59,52,61)(45,64)(46,63)(49,53)(57,58,62,60);(1,5)(2,10,11,22)(3,15)(4,19)(6,23)(7,24)(8,29,13,42)(9,32,33,36)(12,37,38,52)(14,43)(16,45)(17,46)(18,49)(20,50)(21,51)(25,53)(26,54)(27,35)(28,55,41,56)(30,57,39,62)(31,58,48,60)(34,59,40,61)(44,63)(47,64)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=e,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,d^c=de,f^c=fi,g^c=gi,h^c=hi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367656);