Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454977 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
871 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 72 points:
(1,2,4,5)(3,6,7,8)(9,10,16,36,14,20,15,21)(11,22,34,40,25,17,26,47)(12,18,35,48,29,43,30,44)(13,31,37,46,32,19,33,60)(23,49,55,41,52,38,27,56)(24,39,61,70,53,50,54,65)(28,42,62,69,58,45,59,67)(51,64,72,66,71,63,57,68);(2,5)(6,8)(9,11,14,25)(10,17,20,22)(12,27,29,55)(13,24,32,53)(15,34,16,26)(18,41,43,56)(19,39,31,50)(21,47,36,40)(23,35,52,30)(28,57,58,72)(33,61,37,54)(38,48,49,44)(42,66,45,68)(46,70,60,65)(51,62,71,59)(63,69,64,67);(1,3,4,7)(2,6,5,8)(9,12,14,29)(10,18,20,43)(11,23,25,52)(13,28,32,58)(15,35,16,30)(17,38,22,49)(19,45,31,42)(21,48,36,44)(24,51,53,71)(26,55,34,27)(33,62,37,59)(39,64,50,63)(40,56,47,41)(46,67,60,69)(54,72,61,57)(65,66,70,68);(3,7)(6,8)(9,13)(10,19,20,31)(11,24)(12,28)(14,32)(15,33)(16,37)(17,39,22,50)(18,42,43,45)(21,46,36,60)(23,51)(25,53)(26,54)(27,57)(29,58)(30,59)(34,61)(35,62)(38,63,49,64)(40,65,47,70)(41,66,56,68)(44,67,48,69)(52,71)(55,72)Polycyclic presentation:
2^9|a^2=efg,b^2=i,c^2=ef,d^2=fi,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,d^c=de,g^c=gi,h^c=hi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 370732);