Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1215 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)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26,29,33)(27,35,38,31)(28,39,41,32)(30,34,43,50)(36,46,54,53)(37,51,44,47)(40,48,55,56)(42,52,45,49);(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,27,29,38)(26,31,33,35)(28,40,41,55)(30,44,43,37)(32,48,39,56)(34,51,50,47)(36,45,54,42)(46,52,53,49);(2,5)(6,8)(25,28,29,41)(26,32,33,39)(27,36,38,54)(30,45,43,42)(31,46,35,53)(34,52,50,49)(37,55,44,40)(47,56,51,48);(1,3,4,7)(2,6,5,8)(10,15)(14,17)(16,23)(22,24)(26,33)(28,41)(31,35)(34,50)(36,54)(40,55)(42,45)(47,51);(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(27,37)(28,41)(30,43)(31,47)(32,39)(34,50)(35,51)(36,40)(38,44)(46,48)(53,56)(54,55)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6611511);