Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,4,3)(5,6)(7,13)(8,12)(9,11)(10,14)(15,19)(16,18)(17,20)(21,22)(23,28)(24,35)(25,30)(26,40)(27,31)(29,37)(32,46)(33,42)(34,44)(36,45)(38,49)(39,43)(41,47)(48,51)(50,52);(1,3,4,2)(5,7)(6,10)(8,15)(9,14)(11,13)(12,17)(16,20)(18,19)(21,23,27,34)(22,28,31,44)(24,32,38,48)(25,39,26,33)(29,41,45,52)(30,43,40,42)(35,46,49,51)(36,50,37,47);(7,14)(10,13)(15,20)(17,19)(21,24,25,36,27,38,26,37)(22,29,40,49,31,45,30,35)(23,32,39,50,34,48,33,47)(28,41,42,51,44,52,43,46);(6,11)(10,13)(12,18)(17,19)(21,25,27,26)(22,30,31,40)(23,33,34,39)(24,36,38,37)(28,42,44,43)(29,35,45,49)(32,47,48,50)(41,51,52,46);(5,8,9,16)(6,12,11,18)(7,15,14,20)(10,17,13,19)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=gi,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,d^b=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6283717);