Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7309080 |
| 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 | |
1146 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)(25,26,32,38,31,37,30,36)(27,39,41,53,42,33,47,50)(28,35,43,51,29,44,46,34)(40,49,56,48,45,52,55,54);(2,6)(3,4)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,27)(26,33)(28,40)(29,45)(30,47)(31,42)(32,41)(34,48)(35,52)(36,53)(37,39)(38,50)(43,56)(44,49)(46,55)(51,54);(1,2)(3,8)(4,7)(5,6)(25,28,31,29)(26,34,37,51)(27,40,42,45)(30,43,32,46)(33,48,39,54)(35,38,44,36)(41,55,47,56)(49,53,52,50);(1,3,5,4)(2,7,6,8)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,29,31,28)(26,35,37,44)(27,40,42,45)(30,43,32,46)(33,49,39,52)(34,38,51,36)(41,56,47,55)(48,53,54,50);(1,4,5,3)(2,7,6,8)(10,16)(14,17)(15,24)(22,23)(25,30,31,32)(26,36,37,38)(27,41,42,47)(28,43,29,46)(33,50,39,53)(34,35,51,44)(40,55,45,56)(48,49,54,52)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=i,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,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, 7397952);