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 | #45092 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1452 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)(27,33)(28,42)(29,35)(30,36)(31,46)(32,37)(34,43)(38,53)(39,48)(40,49)(41,51)(44,52)(45,50)(47,55)(54,56);(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)(26,33)(28,38)(29,39)(30,45)(31,40)(32,41)(34,47)(35,48)(36,50)(37,51)(42,53)(43,55)(44,54)(46,49)(52,56);(1,3)(2,5)(4,8)(6,7)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28,32,29)(26,34,37,52)(27,38,41,39)(30,43,31,44)(33,47,51,56)(35,46,42,36)(40,54,45,55)(48,49,53,50);(2,6)(3,8)(11,19)(14,17)(18,24)(21,23)(25,29,32,28)(26,35,37,42)(27,39,41,38)(30,43,31,44)(33,48,51,53)(34,46,52,36)(40,54,45,55)(47,49,56,50);(2,6)(5,7)(10,16)(14,17)(15,22)(21,23)(25,30,32,31)(26,36,37,46)(27,40,41,45)(28,43,29,44)(33,49,51,50)(34,35,52,42)(38,54,39,55)(47,53,56,48)Polycyclic presentation:
2^9|c^2=fi,d^2=i,e^2=i,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,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, 7209047);