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 | |
1663 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)(21,22,27,41,26,28,25,31)(23,32,34,29,35,44,36,42)(24,37,39,46,38,30,40,45)(33,48,51,52,49,47,50,43);(1,2,3,4)(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(22,28)(24,38)(29,42)(31,41)(32,44)(33,49)(39,40)(50,51);(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,23,26,35)(22,29,28,42)(24,33,38,49)(25,34,27,36)(30,43,37,52)(31,44,41,32)(39,50,40,51)(45,47,46,48);(2,4)(7,14)(10,13)(15,20)(18,19)(21,24)(22,30)(23,33)(25,40)(26,38)(27,39)(28,37)(29,43)(31,46)(32,47)(34,51)(35,49)(36,50)(41,45)(42,52)(44,48);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(21,25,26,27)(22,31,28,41)(23,34,35,36)(24,39,38,40)(29,44,42,32)(30,45,37,46)(33,50,49,51)(43,47,52,48)Polycyclic presentation:
2^9|a^2=gi,b^2=fhi,c^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7089746);