Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6885129 |
| 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 | |
1406 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)(25,26,30,34)(27,37,39,32)(28,41,42,50)(29,36,31,35)(33,45,48,43)(38,52,54,56)(40,47,44,49)(46,55,51,53);(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)(30,39)(31,40)(33,46)(34,37)(35,49)(36,47)(41,56)(42,54)(43,55)(45,53)(48,51)(50,52);(2,5)(6,8)(12,21)(15,20)(18,24)(22,23)(25,28,30,42)(26,33,34,48)(27,38,39,54)(29,43,31,45)(32,46,37,51)(35,41,36,50)(40,53,44,55)(47,52,49,56);(1,3,4,7)(2,6,5,8)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(26,34)(27,39)(28,42)(35,36)(40,44)(43,45)(46,51)(52,56);(11,19)(14,17)(18,24)(22,23)(25,29,30,31)(26,35,34,36)(27,40,39,44)(28,43,42,45)(32,47,37,49)(33,41,48,50)(38,53,54,55)(46,52,51,56)Polycyclic presentation:
2^9|a^2=fh,b^2=fi,c^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6824404);