Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1217 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)(3,5)(4,6)(7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26)(27,33)(28,41)(29,35)(30,45)(31,37)(32,38)(34,43)(36,47)(39,55)(40,50)(42,51)(44,54)(46,53)(48,52)(49,56);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27,31,40)(26,33,37,50)(28,32,43,29)(30,42,47,46)(34,35,41,38)(36,53,45,51)(39,44,56,48)(49,52,55,54);(2,6)(5,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28,32,40,31,43,29,27)(26,34,38,33,37,41,35,50)(30,46,48,39,47,42,44,56)(36,53,54,55,45,51,52,49);(10,16)(14,17)(15,22)(21,23)(25,29,31,32)(26,35,37,38)(27,28,40,43)(30,44,47,48)(33,41,50,34)(36,52,45,54)(39,42,56,46)(49,53,55,51);(11,19)(14,17)(18,24)(21,23)(25,30,29,44,31,47,32,48)(26,36,35,52,37,45,38,54)(27,39,43,46,40,56,28,42)(33,49,34,51,50,55,41,53)Polycyclic presentation:
2^9|b^2=g,c^2=fh,d^2=i,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6719662);