Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884841 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1416 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,5)(3,7)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26)(27,37)(28,41)(29,35)(30,36)(31,46)(32,40)(33,42)(34,43)(38,48)(39,50)(44,49)(45,51)(47,54)(52,56)(53,55);(1,2,4,5)(3,6,7,8)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,27)(26,32)(28,38)(29,44)(30,40)(31,39)(33,47)(34,48)(35,50)(36,37)(41,53)(42,56)(43,55)(45,54)(46,49)(51,52);(2,5)(6,8)(12,21)(15,20)(18,24)(22,23)(25,28,30,43)(26,33,36,51)(27,38,40,55)(29,42,31,45)(32,47,37,52)(34,46,41,35)(39,54,44,56)(48,49,53,50);(1,3,4,7)(2,6,5,8)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,28)(26,34)(27,38)(29,45)(30,43)(31,42)(32,48)(33,35)(36,41)(37,53)(39,56)(40,55)(44,54)(46,51)(47,50)(49,52);(11,19)(14,17)(18,24)(22,23)(25,29,30,31)(26,35,36,46)(27,39,40,44)(28,42,43,45)(32,49,37,50)(33,34,51,41)(38,54,55,56)(47,53,52,48)Polycyclic presentation:
2^9|a^2=fhi,b^2=fi,c^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,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, 6824324);