Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219532 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2328 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
934 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)(3,5)(4,7)(6,8)(9,10,15,21,16,22,14,20)(11,23,27,17,28,40,26,41)(12,29,31,18,33,44,32,45)(13,19,36,48,37,49,35,47)(24,38,52,50,54,42,53,57)(25,46,55,39,56,65,34,63)(30,58,60,43,62,68,61,69)(51,64,59,70,72,66,71,67);(1,3)(2,4)(5,8)(6,7)(9,11,16,28)(10,17,22,41)(12,24,33,54)(13,34,37,55)(14,27,15,26)(18,42,45,38)(19,46,49,65)(20,40,21,23)(25,36,56,35)(29,57,44,50)(30,59,62,71)(31,53,32,52)(39,48,63,47)(43,67,69,70)(51,61,72,60)(58,64,68,66);(1,4,6,5)(2,7,8,3)(9,12)(10,18,22,45)(11,24)(13,30)(14,31)(15,32)(16,33)(17,38,41,42)(19,43,49,69)(20,44,21,29)(23,50,40,57)(25,51)(26,52)(27,53)(28,54)(34,59)(35,60)(36,61)(37,62)(39,64,63,66)(46,70,65,67)(47,68,48,58)(55,71)(56,72);(1,5,6,4)(2,7,8,3)(9,13,15,36,16,37,14,35)(10,19,21,48,22,49,20,47)(11,25,27,55,28,56,26,34)(12,30,32,61,33,62,31,60)(17,39,40,65,41,63,23,46)(18,43,29,58,45,69,44,68)(24,51,53,71,54,72,52,59)(38,64,57,67,42,66,50,70)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=efg,d^2=ei,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293725);