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, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1151 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,31,37,32,38,30,36)(27,33,42,50,43,51,41,49)(28,44,45,52,29,35,46,34)(39,53,54,56,40,48,55,47);(2,6)(3,4)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,27,32,43)(26,33,38,51)(28,39,29,40)(30,41,31,42)(34,47,52,56)(35,48,44,53)(36,49,37,50)(45,54,46,55);(1,2)(3,8)(4,7)(5,6)(25,28,32,29)(26,34,38,52)(27,39,43,40)(30,45,31,46)(33,47,51,56)(35,36,44,37)(41,54,42,55)(48,49,53,50);(1,3,5,4)(2,7,6,8)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,29,32,28)(26,35,38,44)(27,40,43,39)(30,45,31,46)(33,48,51,53)(34,36,52,37)(41,54,42,55)(47,49,56,50);(1,4,5,3)(2,7,6,8)(10,16)(14,17)(15,24)(22,23)(25,30,32,31)(26,36,38,37)(27,41,43,42)(28,45,29,46)(33,49,51,50)(34,44,52,35)(39,54,40,55)(47,53,56,48)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=i,d^2=fhi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,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, 7397506);