Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1280 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,26)(27,37)(28,34)(29,35)(30,36)(31,33)(32,40)(38,51)(39,45)(41,47)(42,48)(43,49)(44,50)(46,53)(52,56)(54,55);(2,5)(6,8)(9,11,14,21)(10,15,18,24)(12,20,13,19)(16,23,17,22)(25,27,30,40)(26,32,36,37)(28,38,43,53)(29,39,31,41)(33,45,35,47)(34,46,49,51)(42,52,44,54)(48,55,50,56);(9,12,14,13)(10,16,18,17)(11,19,21,20)(15,22,24,23)(26,33)(29,31)(32,45)(34,48)(35,36)(37,47)(39,41)(42,44)(46,55)(49,50)(51,56)(52,54);(9,13,14,12)(10,17,18,16)(11,20,21,19)(15,23,24,22)(25,28,30,43)(26,34,36,49)(27,38,40,53)(29,44,31,42)(32,46,37,51)(33,48,35,50)(39,54,41,52)(45,55,47,56);(1,3,4,7)(2,6,5,8)(10,18)(15,24)(16,17)(22,23)(25,29,30,31)(26,35,36,33)(27,39,40,41)(28,42,43,44)(32,47,37,45)(34,48,49,50)(38,52,53,54)(46,55,51,56)Polycyclic presentation:
2^9|a^2=fi,b^2=hi,c^2=h,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^c=ei,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6397403);