Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #405621 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14915 |
| 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 | |
1285 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14)(3,15,19,34)(4,21,26,37)(5,27,30,39)(6,13,7,12)(9,20,33,18)(10,24,38,25)(11,28,40,29)(16,31,22,35)(17,32,48,57)(23,36,52,60)(41,44,56,45)(42,46,54,47)(43,61,50,64)(49,53,59,51)(55,63,58,62);(1,3,7,20,8,19,6,18)(2,9,13,34,14,33,12,15)(4,22,24,45,26,16,25,44)(5,17,29,47,30,48,28,46)(10,35,37,56,38,31,21,41)(11,32,27,54,40,57,39,42)(23,50,51,63,52,43,53,62)(36,58,59,64,60,55,49,61);(1,4)(2,10)(3,16)(5,23)(6,24)(7,25)(8,26)(9,31)(11,36)(12,37)(13,21)(14,38)(15,41)(17,43)(18,44)(19,22)(20,45)(27,49)(28,51)(29,53)(30,52)(32,55)(33,35)(34,56)(39,59)(40,60)(42,61)(46,62)(47,63)(48,50)(54,64)(57,58);(1,5,6,28,8,30,7,29)(2,11,12,39,14,40,13,27)(3,17,18,46,19,48,20,47)(4,23,24,51,26,52,25,53)(9,32,15,42,33,57,34,54)(10,36,37,59,38,60,21,49)(16,43,44,62,22,50,45,63)(31,55,41,61,35,58,56,64)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,f^c=fi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312818);