Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7300916 |
| 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 | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1425 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)(25,26,30,36)(27,38,41,32)(28,34,44,51)(29,37,31,35)(33,45,50,43)(39,53,55,47)(40,48,42,49)(46,54,52,56);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27,30,41)(26,32,36,38)(28,39,44,55)(29,40,31,42)(33,46,50,52)(34,47,51,53)(35,48,37,49)(43,54,45,56);(1,3)(2,5)(4,8)(6,7)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28)(26,33)(27,39)(29,43)(30,44)(31,45)(32,46)(34,35)(36,50)(37,51)(38,52)(40,54)(41,55)(42,56)(47,48)(49,53);(2,6)(3,8)(11,19)(14,17)(18,24)(21,23)(25,28)(26,34)(27,39)(29,45)(30,44)(31,43)(32,47)(33,35)(36,51)(37,50)(38,53)(40,56)(41,55)(42,54)(46,48)(49,52);(2,6)(5,7)(10,16)(14,17)(15,22)(21,23)(25,29,30,31)(26,35,36,37)(27,40,41,42)(28,43,44,45)(32,48,38,49)(33,34,50,51)(39,54,55,56)(46,47,52,53)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,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, 7209197);