Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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 | #45101 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1723 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)(21,22,27,28,26,32,25,31)(23,33,40,45,35,29,36,42)(24,30,41,43,39,48,37,47)(34,46,50,52,38,44,49,51);(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(22,28)(24,37)(25,27)(29,42)(31,32)(33,45)(34,49)(36,40)(38,50)(39,41)(43,47)(51,52);(2,4)(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,23)(22,29)(24,38)(25,36)(26,35)(27,40)(28,42)(30,46)(31,45)(32,33)(34,39)(37,50)(41,49)(43,52)(44,48)(47,51);(7,14)(10,13)(15,20)(18,19)(21,24,27,41,26,39,25,37)(22,30,28,43,32,48,31,47)(23,34,40,50,35,38,36,49)(29,44,42,51,33,46,45,52);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(23,35)(29,33)(34,38)(36,40)(42,45)(44,46)(49,50)(51,52)Polycyclic presentation:
2^9|a^2=fi,b^2=h,d^2=fi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7088126);