Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418508 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2675 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 72 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,23)(12,29)(13,35)(14,21)(15,20)(16,22)(17,26)(18,33)(19,39)(24,40)(25,47)(27,43)(28,42)(30,57)(31,59)(32,46)(34,45)(36,41)(37,48)(38,49)(44,61)(50,53)(51,67)(52,68)(54,69)(55,64)(56,63)(58,71)(60,70)(62,65)(66,72);(1,3,5,8)(2,6,4,7)(9,11,16,28)(10,17,22,43)(12,30,34,24)(13,36,39,54)(14,27,15,26)(18,40,46,57)(19,47,35,64)(20,42,21,23)(25,38,55,37)(29,50,45,68)(31,60,63,66)(32,52,33,53)(41,49,69,48)(44,70,65,72)(51,61,58,62)(56,71,59,67);(1,4,5,2)(3,7,8,6)(9,12,15,33,16,34,14,32)(10,18,21,29,22,46,20,45)(11,24,27,53,28,30,26,52)(13,31,38,62,39,63,37,61)(17,40,42,68,43,57,23,50)(19,44,49,56,35,65,48,59)(25,51,54,60,55,58,36,66)(41,67,64,72,69,71,47,70);(3,8)(6,7)(9,13,15,38,16,39,14,37)(10,19,21,49,22,35,20,48)(11,25,27,54,28,55,26,36)(12,31,33,62,34,63,32,61)(17,41,42,64,43,69,23,47)(18,44,29,56,46,65,45,59)(24,51,53,60,30,58,52,66)(40,67,68,72,57,71,50,70)Polycyclic presentation:
2^9|a^2=ef,b^2=efi,c^2=ei,d^2=f,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, 337434);