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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1656 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)(8,16)(11,17)(15,20)(18,19)(21,22,26,31,27,32,25,28)(23,29,35,44,33,46,41,42)(24,36,37,43,40,30,39,48)(34,47,50,51,38,45,49,52);(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(22,28)(23,33)(24,37)(25,26)(29,44)(31,32)(34,49)(38,50)(39,40)(42,46)(43,48)(45,47);(2,4)(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,23)(22,29)(24,38)(25,41)(26,35)(27,33)(28,42)(30,47)(31,44)(32,46)(34,40)(36,45)(37,49)(39,50)(43,52)(48,51);(7,14)(10,13)(15,20)(18,19)(21,24,25,39,27,40,26,37)(22,30,28,43,32,36,31,48)(23,34,41,49,33,38,35,50)(29,45,42,51,46,47,44,52);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(21,25,27,26)(22,31,32,28)(23,35,33,41)(24,39,40,37)(29,42,46,44)(30,48,36,43)(34,50,38,49)(45,51,47,52)Polycyclic presentation:
2^9|a^2=fg,b^2=hi,d^2=fi,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, 7090811);