Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884969 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1389 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,16)(12,21)(13,17)(14,18)(15,20)(19,23)(22,24)(25,26,31,37)(27,39,43,33)(28,35,29,46)(30,38,32,36)(34,47,53,45)(40,54,41,49)(42,50,44,51)(48,55,52,56);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,18)(17,23)(21,24)(25,27,31,43)(26,33,37,39)(28,41,29,40)(30,42,32,44)(34,52,53,48)(35,54,46,49)(36,50,38,51)(45,56,47,55);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,22)(13,19)(14,20)(15,18)(17,23)(21,24)(25,28,31,29)(26,34,37,53)(27,40,43,41)(30,45,32,47)(33,48,39,52)(35,38,46,36)(42,55,44,56)(49,51,54,50);(3,8)(5,7)(9,12,14,13)(10,17,18,21)(11,19,20,22)(15,23,16,24)(25,29,31,28)(26,35,37,46)(27,41,43,40)(30,45,32,47)(33,49,39,54)(34,38,53,36)(42,55,44,56)(48,51,52,50);(2,6)(5,7)(9,13,14,12)(10,17,18,21)(11,19,20,22)(15,24,16,23)(25,30,31,32)(26,36,37,38)(27,42,43,44)(28,45,29,47)(33,50,39,51)(34,35,53,46)(40,55,41,56)(48,49,52,54)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6798689);