Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26)(27,36)(28,40)(29,34)(30,32)(31,38)(33,42)(35,44)(37,46)(39,47)(41,49)(43,50)(45,52)(48,54)(51,55)(53,56);(1,2)(3,7)(4,5)(6,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27,29,38)(26,31,34,36)(28,41,42,37)(30,39,44,52)(32,45,35,47)(33,49,40,46)(43,53,54,51)(48,56,50,55);(2,5)(6,7)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(26,32)(30,44)(31,45)(33,48)(34,35)(36,47)(39,52)(40,50)(43,54)(46,55)(49,56)(51,53);(10,16)(14,17)(15,22)(21,23)(25,28,29,42)(26,33,34,40)(27,37,38,41)(30,43,44,54)(31,46,36,49)(32,48,35,50)(39,51,52,53)(45,55,47,56);(1,3,4,8)(2,6,5,7)(11,19)(14,17)(18,24)(21,23)(27,38)(28,42)(31,36)(33,40)(39,52)(43,54)(45,47)(48,50)Polycyclic presentation:
2^9|b^2=i,c^2=fi,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6698862);