Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884969 |
| 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 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1400 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)(25,26,31,37)(27,32,42,50)(28,34,44,53)(29,36,30,35)(33,45,51,43)(38,52,39,47)(40,49,41,48)(46,55,56,54);(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)(29,40)(30,41)(31,42)(33,46)(34,52)(35,48)(36,49)(37,50)(39,44)(43,54)(45,55)(47,53)(51,56);(2,5)(6,8)(12,21)(15,20)(18,24)(22,23)(25,28)(26,33)(27,38)(29,43)(30,45)(31,44)(32,46)(34,35)(36,53)(37,51)(39,42)(40,54)(41,55)(47,49)(48,52)(50,56);(1,3,4,7)(2,6,5,8)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,28)(26,34)(27,39)(29,45)(30,43)(31,44)(32,47)(33,35)(36,51)(37,53)(38,42)(40,54)(41,55)(46,49)(48,56)(50,52);(11,19)(14,17)(18,24)(22,23)(25,29,31,30)(26,35,37,36)(27,40,42,41)(28,43,44,45)(32,48,50,49)(33,34,51,53)(38,54,39,55)(46,52,56,47)Polycyclic presentation:
2^9|a^2=fhi,b^2=f,d^2=f,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=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6824470);