Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395190 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2431 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14905 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1082 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)(3,13)(4,18)(5,22)(6,25)(7,12)(8,16)(9,21)(10,24)(11,27)(14,39)(15,43)(17,38)(19,32)(20,33)(23,34)(26,35)(28,41)(29,56)(30,47)(31,46)(36,42)(37,45)(40,60)(44,51)(48,50)(49,59)(52,54)(53,58)(55,64)(57,62)(61,63);(1,3,6,16,7,17,27,47)(2,8,11,13,12,30,25,38)(4,14,26,46,21,42,20,41)(5,15,23,44,24,45,54,56)(9,31,35,39,18,28,33,36)(10,29,34,37,22,51,52,43)(19,40,53,64,50,63,49,62)(32,57,59,61,48,55,58,60);(1,4,6,26,7,21,27,20)(2,9,11,35,12,18,25,33)(3,14,16,46,17,42,47,41)(5,19,23,53,24,50,54,49)(8,28,13,36,30,31,38,39)(10,32,34,59,22,48,52,58)(15,40,44,64,45,63,56,62)(29,55,37,60,51,57,43,61);(1,5)(2,10,12,22)(3,15)(4,19)(6,23)(7,24)(8,29,30,51)(9,32,18,48)(11,34,25,52)(13,37,38,43)(14,40)(16,44)(17,45)(20,49)(21,50)(26,53)(27,54)(28,55,31,57)(33,58,35,59)(36,60,39,61)(41,62)(42,63)(46,64)(47,56)Polycyclic presentation:
2^9|b^2=ef,c^2=ef,d^2=g,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^c=ei,f^c=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312073);