Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7200022 |
| 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 | #36015 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1388 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,29,34)(27,36,37,31)(28,40,41,33)(30,32,43,35)(38,45,50,49)(39,44,51,46)(42,48,53,47)(52,55,56,54);(2,5)(3,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,29,37)(26,31,34,36)(28,38,41,50)(30,39,43,51)(32,46,35,44)(33,45,40,49)(42,52,53,56)(47,55,48,54);(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(26,32)(27,37)(30,43)(31,44)(33,47)(34,35)(36,46)(38,50)(40,48)(42,53)(45,54)(49,55);(1,3,4,7)(2,6,5,8)(10,16)(11,18)(15,24)(17,19)(25,28,29,41)(26,33,34,40)(27,38,37,50)(30,42,43,53)(31,45,36,49)(32,47,35,48)(39,52,51,56)(44,54,46,55);(2,5)(6,8)(11,18)(14,22)(17,19)(21,23)(26,34)(27,37)(28,41)(32,35)(39,51)(42,53)(45,49)(54,55)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=hi,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,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, 7119017);