Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 | |
1649 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)(26,32)(27,33)(28,31)(29,37)(30,35)(36,48)(39,51)(41,52)(42,47)(43,46)(44,50)(45,49);(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)(32,45)(33,34)(36,41)(38,40)(42,49)(43,50)(44,52)(46,48)(47,51);(2,4)(21,24,25,37,27,35,28,23)(22,30,40,34,33,38,31,29)(26,39,41,50,42,49,43,36)(32,45,52,48,47,51,46,44);(6,11)(10,13)(12,19)(18,20)(21,25,27,28)(22,31,33,40)(23,35,37,24)(26,41,42,43)(29,30,34,38)(32,46,47,52)(36,49,50,39)(44,45,48,51);(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(21,26,28,43,27,42,25,41)(22,32,31,46,33,47,40,52)(23,36,35,49,37,50,24,39)(29,44,38,51,34,48,30,45)Polycyclic presentation:
2^9|b^2=fi,c^2=hi,d^2=i,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6716709);