Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7300980 |
| 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×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1353 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)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(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);(1,2)(3,7)(4,6)(5,8)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(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);(1,3)(2,5)(4,8)(6,7)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(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)(3,8)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28,30,39)(26,33,34,48)(27,36,38,49)(29,40,41,52)(31,43,44,55)(32,46,35,47)(37,50,51,56)(42,53,45,54);(2,6)(5,7)(10,18)(15,19)(16,17)(23,24)(26,34)(28,39)(31,44)(32,35)(36,49)(40,52)(42,45)(50,56)Polycyclic presentation:
2^9|b^2=fi,c^2=f,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=efi,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,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, 7211786);