Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469971 |
| 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 | |
1389 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)(28,40)(29,34)(30,44)(31,35)(32,38)(33,42)(37,51)(39,47)(41,55)(43,49)(45,48)(46,56)(50,53)(52,54);(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,31,39)(26,32,35,48)(28,37,42,53)(29,38,30,45)(33,46,40,54)(34,47,44,36)(41,52,43,56)(49,51,55,50);(1,3,5,4)(2,6,7,8)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,28,31,42)(26,33,35,40)(27,37,39,53)(29,41,30,43)(32,46,48,54)(34,49,44,55)(36,50,47,51)(38,52,45,56);(11,18)(14,22)(17,19)(21,23)(25,29,31,30)(26,34,35,44)(27,38,39,45)(28,41,42,43)(32,47,48,36)(33,49,40,55)(37,52,53,56)(46,51,54,50);(1,4,5,3)(2,6,7,8)(10,16)(14,22)(15,24)(21,23)(28,42)(33,40)(37,53)(41,43)(46,54)(49,55)(50,51)(52,56)Polycyclic presentation:
2^9|b^2=i,c^2=gh,d^2=i,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6388159);