Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418508 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2675 |
| 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 | |
1610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2,3,4)(5,6)(7,19)(8,25)(9,31)(10,17)(11,16)(12,18)(13,22)(14,29)(15,35)(20,36)(21,43)(23,39)(24,38)(26,53)(27,55)(28,42)(30,41)(32,37)(33,44)(34,45)(40,57)(46,49)(47,63)(48,64)(50,65)(51,60)(52,59)(54,67)(56,66)(58,61)(62,68);(5,7)(6,13)(8,26)(9,32)(10,23)(11,22)(12,24)(14,36)(15,43)(16,38)(17,19)(18,39)(20,30)(21,33)(25,46)(27,56)(28,48)(29,49)(31,60)(34,51)(35,50)(37,44)(40,66)(41,64)(42,53)(45,65)(47,58)(52,67)(54,57)(55,63)(59,62)(61,68);(2,4)(5,8,11,29,12,30,10,28)(6,14,17,25,18,42,16,41)(7,20,23,49,24,26,22,48)(9,27,34,58,35,59,33,57)(13,36,38,64,39,53,19,46)(15,40,45,52,31,61,44,55)(21,47,50,56,51,54,32,62)(37,63,60,68,65,67,43,66);(5,9,10,33,12,35,11,34)(6,15,16,44,18,31,17,45)(7,21,22,32,24,51,23,50)(8,27,28,57,30,59,29,58)(13,37,19,43,39,65,38,60)(14,40,41,55,42,61,25,52)(20,47,48,62,26,54,49,56)(36,63,46,66,53,67,64,68)Polycyclic presentation:
2^9|a^2=ef,c^2=ei,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,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, 327601);