Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470374 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1305 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26)(27,36)(28,33)(29,34)(30,44)(31,35)(32,39)(37,49)(38,46)(40,50)(41,47)(42,48)(43,54)(45,53)(51,55)(52,56);(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,30,39,31,40,29,38)(26,32,34,36,35,46,44,50)(28,37,43,53,42,52,41,51)(33,45,47,49,48,55,54,56);(1,3,5,4)(2,6,7,8)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,28)(26,33)(27,37)(29,41)(30,43)(31,42)(32,45)(34,47)(35,48)(36,49)(38,51)(39,53)(40,52)(44,54)(46,55)(50,56);(11,18)(14,22)(17,19)(21,23)(25,29,31,30)(26,34,35,44)(27,38,40,39)(28,41,42,43)(32,36,46,50)(33,47,48,54)(37,51,52,53)(45,49,55,56);(1,4,5,3)(2,6,7,8)(10,16)(14,22)(15,24)(21,23)(28,42)(33,48)(37,52)(41,43)(45,55)(47,54)(49,56)(51,53)Polycyclic presentation:
2^9|b^2=f,c^2=gh,d^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6388398);