Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500535 |
| 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×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1271 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,36)(28,33)(29,34)(30,45)(31,35)(32,39)(37,48)(38,53)(40,56)(41,50)(42,52)(43,46)(44,51)(47,55)(49,54);(2,5)(6,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(27,37)(32,46)(36,48)(38,40)(39,43)(47,49)(53,56)(54,55);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,30,39,31,37,29,43)(26,32,34,36,35,46,45,48)(28,40,41,54,42,38,44,55)(33,49,51,56,52,47,50,53);(1,3,4,7)(2,6,5,8)(12,20)(15,24)(17,19)(21,23)(25,28,31,42)(26,33,35,52)(27,38,37,40)(29,44,30,41)(32,47,46,49)(34,51,45,50)(36,53,48,56)(39,55,43,54);(11,18)(14,22)(17,19)(21,23)(25,29,31,30)(26,34,35,45)(27,39,37,43)(28,41,42,44)(32,48,46,36)(33,50,52,51)(38,54,40,55)(47,53,49,56)Polycyclic presentation:
2^9|a^2=hi,c^2=g,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6418221);