Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | #2535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1712 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(8,16)(11,17)(15,20)(18,19)(21,22,27,28,26,32,25,31)(23,29,39,41,34,44,35,45)(24,36,40,46,38,30,37,42)(33,47,50,52,49,43,48,51);(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(22,28)(24,37)(25,27)(29,41)(31,32)(33,48)(35,39)(38,40)(42,46)(44,45)(49,50)(51,52);(1,2,3,4)(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,23)(22,29)(24,33)(25,35)(26,34)(27,39)(28,41)(30,43)(31,45)(32,44)(36,47)(37,48)(38,49)(40,50)(42,51)(46,52);(7,14)(10,13)(15,20)(18,19)(21,24,25,37,26,38,27,40)(22,30,31,46,32,36,28,42)(23,33,35,48,34,49,39,50)(29,43,45,52,44,47,41,51);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(23,34)(29,44)(33,49)(35,39)(41,45)(43,47)(48,50)(51,52)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=g,d^2=f,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7093870);