Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1639 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)(3,4)(5,6)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22)(23,34)(24,38)(25,40)(26,42)(27,33)(28,31)(29,37)(30,35)(32,44)(36,46)(39,47)(41,48)(43,49)(45,52)(50,51);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23)(22,29)(24,28)(25,35)(26,39)(27,37)(30,31)(32,47)(33,34)(36,41)(38,40)(42,49)(43,44)(45,51)(46,52)(48,50);(2,4)(21,24,25,37,27,35,28,23)(22,30,40,34,33,38,31,29)(26,43,41,36,44,39,45,51)(32,49,52,46,42,47,48,50);(6,11)(10,13)(12,19)(18,20)(21,25,27,28)(22,31,33,40)(23,35,37,24)(26,41,44,45)(29,30,34,38)(32,48,42,52)(36,43,51,39)(46,47,50,49);(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(21,26,25,41,27,44,28,45)(22,32,40,52,33,42,31,48)(23,36,24,39,37,51,35,43)(29,46,30,47,34,50,38,49)Polycyclic presentation:
2^9|b^2=fi,c^2=hi,d^2=i,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6716929);