Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2428 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1467 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(2,4)(5,6,10,16,11,17,12,34)(7,18,21,38,22,39,33,13)(8,23,26,54,27,14,28,42)(9,29,31,15,32,44,35,45)(19,47,49,36,24,46,56,40)(20,50,30,63,51,37,52,43)(25,57,59,53,60,62,61,41)(48,65,64,68,55,66,58,67);(5,7,11,22)(6,13,17,38)(8,24,27,19)(9,30,32,52)(10,33,12,21)(14,40,23,36)(15,43,45,63)(16,39,34,18)(20,35,51,31)(25,58,60,64)(26,49,28,56)(29,50,44,37)(41,67,53,68)(42,46,54,47)(48,59,55,61)(57,66,62,65);(1,2,3,4)(5,8,12,28,11,27,10,26)(6,14,34,54,17,23,16,42)(7,19,33,56,22,24,21,49)(9,25,35,61,32,60,31,59)(13,36,39,47,38,40,18,46)(15,41,29,62,45,53,44,57)(20,48,52,58,51,55,30,64)(37,65,63,67,50,66,43,68);(5,9,12,35,11,32,10,31)(6,15,34,29,17,45,16,44)(7,20,33,52,22,51,21,30)(8,25,28,61,27,60,26,59)(13,37,39,63,38,50,18,43)(14,41,54,62,23,53,42,57)(19,48,56,58,24,55,49,64)(36,65,47,67,40,66,46,68)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=ef,d^2=ei,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 331489);