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 | |
1282 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)(25,26,30,36)(27,37,40,32)(28,34,42,50)(29,35,31,33)(38,51,52,46)(39,45,41,47)(43,48,44,49)(53,56,54,55);(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)(28,38,42,52)(29,39,31,41)(33,45,35,47)(34,46,50,51)(43,53,44,54)(48,55,49,56);(9,12,14,13)(10,16,18,17)(11,19,21,20)(15,22,24,23)(26,33)(28,42)(29,31)(32,45)(34,49)(35,36)(37,47)(38,52)(39,41)(46,56)(48,50)(51,55);(9,13,14,12)(10,17,18,16)(11,20,21,19)(15,23,24,22)(25,28)(26,34)(27,38)(29,44)(30,42)(31,43)(32,46)(33,48)(35,49)(36,50)(37,51)(39,54)(40,52)(41,53)(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)(28,43,42,44)(32,47,37,45)(34,48,50,49)(38,53,52,54)(46,55,51,56)Polycyclic presentation:
2^9|a^2=f,b^2=hi,c^2=h,d^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,d^c=di,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, 6397513);