Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7309060 |
| 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 | #42683 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1419 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)(12,16)(13,17)(15,18)(19,22)(20,23)(21,24)(25,26,31,33,30,36,29,35)(27,32,42,47,37,46,39,44)(28,34,43,48,40,50,41,49)(38,45,53,55,51,56,52,54);(2,5)(3,7)(9,11,13,18)(10,14,17,15)(12,20,21,19)(16,23,24,22)(25,27,30,37)(26,32,36,46)(28,38,40,51)(29,39,31,42)(33,47,35,44)(34,45,50,56)(41,52,43,53)(48,55,49,54);(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,21)(22,24)(26,33)(27,37)(28,40)(29,31)(32,44)(34,49)(35,36)(45,55)(46,47)(48,50)(52,53)(54,56);(1,3,4,7)(2,6,5,8)(11,18)(14,15)(19,20)(22,23)(25,28)(26,34)(27,38)(29,41)(30,40)(31,43)(32,45)(33,48)(35,49)(36,50)(37,51)(39,52)(42,53)(44,54)(46,56)(47,55);(2,5)(6,8)(9,12,13,21)(10,16,17,24)(11,19,18,20)(14,22,15,23)(27,37)(28,40)(32,46)(34,50)(39,42)(41,43)(44,47)(48,49)Polycyclic presentation:
2^9|a^2=gh,b^2=h,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfhi,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, 7394700);