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 | |
1432 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)(25,26,29,34)(27,36,39,31)(28,33,41,51)(30,44,43,35)(32,45,49,42)(37,47,38,50)(40,48,56,53)(46,54,52,55);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27,29,39)(26,31,34,36)(28,37,41,38)(30,40,43,56)(32,46,49,52)(33,50,51,47)(35,48,44,53)(42,54,45,55);(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)(26,32)(27,37)(29,41)(30,45)(31,46)(33,44)(34,49)(35,51)(36,52)(38,39)(40,55)(42,43)(47,48)(50,53)(54,56);(2,6)(3,8)(11,19)(14,17)(18,24)(21,23)(25,28)(26,33)(27,38)(29,41)(30,42)(31,47)(32,44)(34,51)(35,49)(36,50)(37,39)(40,55)(43,45)(46,48)(52,53)(54,56);(2,6)(5,7)(10,16)(14,17)(15,22)(21,23)(26,34)(27,39)(28,41)(35,44)(40,56)(42,45)(46,52)(47,50)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,e^c=ei,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, 7209132);