Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | |
1664 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)(23,33)(24,29)(25,30)(26,41)(27,32)(28,36)(31,34)(35,47)(37,49)(38,48)(39,52)(40,43)(42,46)(44,51)(45,50);(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,34,50,27,35,26,36)(22,28,41,47,32,45,31,33)(24,37,51,40,38,25,39,42)(29,46,52,30,48,43,44,49);(2,4)(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24)(22,29)(23,25)(26,39)(27,38)(28,43)(30,33)(31,44)(32,48)(34,51)(35,37)(36,40)(41,52)(42,50)(45,46)(47,49);(7,14)(10,13)(15,20)(18,19)(21,25,27,37)(22,30,32,49)(23,24,35,38)(26,42,34,40)(28,44,45,52)(29,47,48,33)(31,43,41,46)(36,51,50,39);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(23,35)(24,38)(28,45)(29,48)(33,47)(36,50)(39,51)(44,52)Polycyclic presentation:
2^9|a^2=g,b^2=fh,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7089944);