Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7200042 |
| 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 | |
1434 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(25,26)(27,31)(28,33)(29,32)(30,34)(35,41)(36,43)(37,42)(38,44)(39,48)(40,46)(45,51)(47,52)(49,55)(50,53)(54,56);(1,3)(2,5)(4,7)(6,8)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27,30,38)(26,31,34,44)(28,36,39,49)(29,37,41,51)(32,42,35,45)(33,43,48,55)(40,50,52,56)(46,53,47,54);(3,7)(5,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(26,32)(29,41)(31,42)(33,46)(34,35)(37,51)(40,52)(43,53)(44,45)(47,48)(50,56)(54,55);(2,6)(3,7)(11,19)(14,17)(18,24)(21,23)(25,28)(26,33)(27,36)(29,40)(30,39)(31,43)(32,46)(34,48)(35,47)(37,50)(38,49)(41,52)(42,53)(44,55)(45,54)(51,56);(3,7)(5,8)(10,16)(14,17)(15,22)(21,23)(26,34)(28,39)(31,44)(32,35)(36,49)(40,52)(42,45)(50,56)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=efi,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,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, 7108538);