Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455155 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23438 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1670 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(2,3)(5,6)(7,13)(8,24)(9,29)(10,17)(11,16)(12,18)(14,27)(15,30)(19,45)(20,48)(21,37)(22,36)(23,38)(25,34)(26,40)(28,41)(31,43)(32,42)(33,44)(35,49)(39,47)(46,59)(50,61)(51,60)(52,62)(53,56)(54,63)(55,58)(57,64)(65,66)(67,68);(1,2,4,3)(5,7,10,21,12,23,11,22)(6,13,17,37,18,38,16,36)(8,25,27,47,28,51,32,19)(9,20,30,49,31,50,33,52)(14,39,41,60,42,45,24,34)(15,35,43,61,44,62,29,48)(26,54,55,67,57,66,56,46)(40,63,58,68,64,65,53,59);(2,3)(5,8)(6,14)(7,19)(9,26)(10,32)(11,27)(12,28)(13,34)(15,40)(16,41)(17,24)(18,42)(20,46)(21,51)(22,25)(23,47)(29,58)(30,56)(31,57)(33,55)(35,59)(36,39)(37,45)(38,60)(43,53)(44,64)(48,63)(49,66)(50,67)(52,54)(61,65)(62,68);(1,3,4,2)(5,9,12,31)(6,15,18,44)(7,20,23,50)(8,26,28,57)(10,30,11,33)(13,35,38,62)(14,40,42,64)(16,29,17,43)(19,46,47,67)(21,49,22,52)(24,53,41,58)(25,54,51,66)(27,55,32,56)(34,59,60,68)(36,48,37,61)(39,63,45,65)Polycyclic presentation:
2^9|b^2=g,d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^c=de,f^c=fi,g^c=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 371429);