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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1301 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,7,8,5)(9,10,14,12)(11,17,19,15)(13,20,21,16)(18,22,24,23)(25,26)(27,36)(28,40)(29,34)(30,32)(31,37)(33,42)(35,44)(38,46)(39,45)(41,49)(43,50)(47,51)(48,54)(52,56)(53,55);(1,3,4,8)(2,5,6,7)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,27)(26,31)(28,41)(29,37)(30,39)(32,47)(33,49)(34,36)(35,45)(38,42)(40,46)(43,53)(44,51)(48,55)(50,56)(52,54);(3,8)(5,7)(9,12,14,10)(11,17,19,15)(13,16,21,20)(18,22,24,23)(26,32)(27,37)(30,44)(31,45)(33,48)(34,35)(36,47)(38,41)(40,50)(43,54)(46,55)(49,56);(2,6)(5,7)(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,28,29,42)(26,33,34,40)(27,38,37,41)(30,43,44,54)(31,46,36,49)(32,48,35,50)(39,52,51,53)(45,55,47,56);(11,19)(15,17)(18,24)(22,23)(28,42)(33,40)(38,41)(43,54)(46,49)(48,50)(52,53)(55,56)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=fh,d^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,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, 6861567);