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 | #25553 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1427 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)(27,35)(28,39)(29,31)(30,43)(32,38)(33,40)(34,42)(36,48)(37,51)(41,47)(44,45)(46,56)(49,50)(52,53)(54,55);(1,2,4,5)(3,6,7,8)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(26,31)(27,36)(28,40)(34,43)(38,44)(41,55)(46,53)(49,51);(2,5)(6,8)(12,21)(15,20)(18,24)(22,23)(25,27,29,36)(26,32,31,45)(28,37,40,50)(30,44,42,38)(33,46,39,53)(34,48,43,35)(41,56,55,52)(47,51,54,49);(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)(26,33)(27,37)(29,40)(30,41)(31,39)(32,46)(34,47)(35,49)(36,50)(38,52)(42,55)(43,54)(44,56)(45,53)(48,51);(11,19)(14,17)(18,24)(22,23)(28,40)(33,39)(37,50)(41,55)(46,53)(47,54)(49,51)(52,56)Polycyclic presentation:
2^9|a^2=fh,b^2=fi,c^2=i,d^2=fi,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, 6824435);