Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455561 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2622 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1065 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)(5,6,12,18)(7,19,22,13)(8,24,29,39)(9,15,31,42)(10,17,11,16)(14,28,40,27)(20,47,49,37)(21,46,51,35)(23,36,32,43)(25,45,54,34)(26,55,58,66)(30,44,33,41)(38,59,65,57)(48,67,50,64)(52,62,53,63)(56,61,68,60);(5,7,10,32,12,22,11,23)(6,13,16,43,18,19,17,36)(8,25,27,20,29,54,28,49)(9,21,30,53,31,51,33,52)(14,37,24,34,40,47,39,45)(15,35,41,63,42,46,44,62)(26,56,57,48,58,68,59,50)(38,64,55,60,65,67,66,61);(5,8,12,29)(6,14,18,40)(7,20,22,49)(9,26,31,58)(10,28,11,27)(13,34,19,45)(15,38,42,65)(16,39,17,24)(21,50,51,48)(23,54,32,25)(30,59,33,57)(35,61,46,60)(36,47,43,37)(41,66,44,55)(52,56,53,68)(62,64,63,67);(2,4)(5,9)(6,15)(7,21)(8,26)(10,30)(11,33)(12,31)(13,35)(14,38)(16,41)(17,44)(18,42)(19,46)(20,48)(22,51)(23,52)(24,55)(25,56)(27,57)(28,59)(29,58)(32,53)(34,60)(36,62)(37,64)(39,66)(40,65)(43,63)(45,61)(47,67)(49,50)(54,68)Polycyclic presentation:
2^9|a^2=i,b^2=ef,c^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,d^c=de,f^c=fi,g^c=gi,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 362881);