Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219696 |
| 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)(11,23,27,17,28,42,26,43)(12,29,32,47,33,46,39,18)(13,34,37,49,38,19,36,50)(24,52,30,44,54,40,58,51)(25,55,56,68,57,48,35,41)(31,61,63,45,64,59,66,65)(53,67,60,70,62,71,72,69);(1,3)(2,4)(5,8)(6,7)(9,11,16,28)(10,17,22,43)(12,30,33,58)(13,35,38,56)(14,27,15,26)(18,44,47,51)(19,48,34,55)(20,42,21,23)(24,39,54,32)(25,37,57,36)(29,52,46,40)(31,62,64,53)(41,50,68,49)(45,67,65,71)(59,69,61,70)(60,66,72,63);(1,4,6,5)(2,7,8,3)(9,12,15,39,16,33,14,32)(10,18,20,29,22,47,21,46)(11,24,27,58,28,54,26,30)(13,31,37,66,38,64,36,63)(17,40,23,51,43,52,42,44)(19,45,49,59,34,65,50,61)(25,53,56,72,57,62,35,60)(41,67,48,70,68,71,55,69);(1,5,6,4)(2,7,8,3)(9,13,15,37,16,38,14,36)(10,19,21,50,22,34,20,49)(11,25,27,56,28,57,26,35)(12,31,39,66,33,64,32,63)(17,41,42,55,43,68,23,48)(18,45,46,61,47,65,29,59)(24,53,58,72,54,62,30,60)(40,67,44,69,52,71,51,70)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=efg,d^2=ei,e^2=i,f^2=i,g^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,g^b=gi,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, 293731);