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 | #2619 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1704 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)(5,6,9,12)(7,13,15,10)(8,16,17,11)(14,18,20,19)(21,22,27,26)(23,28,33,42)(24,29,35,37)(25,30,41,40)(31,34,43,49)(32,38,45,52)(36,44,39,46)(47,50,48,51);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,23,27,33)(22,28,26,42)(24,34,35,49)(25,38,41,52)(29,43,37,31)(30,45,40,32)(36,48,39,47)(44,51,46,50);(2,4)(21,24,22,29,27,35,26,37)(23,31,42,49,33,43,28,34)(25,39,30,46,41,36,40,44)(32,48,52,50,45,47,38,51);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(24,35)(29,37)(31,43)(34,49)(36,39)(44,46)(47,48)(50,51);(7,15)(10,13)(14,20)(18,19)(21,25,22,30,27,41,26,40)(23,32,42,52,33,45,28,38)(24,36,29,44,35,39,37,46)(31,47,49,51,43,48,34,50)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=hi,d^2=f,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6715826);