Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469995 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1332 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)(3,7)(4,5)(6,8)(9,10)(11,17)(12,20)(13,15)(14,18)(16,21)(19,22)(23,24)(25,26,29,34,30,33,43,49)(27,35,38,44,36,50,42,31)(28,32,41,47,40,48,55,56)(37,51,52,46,39,53,54,45);(2,5)(6,7)(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24)(25,27,30,36)(26,31,33,44)(28,39,40,37)(29,42,43,38)(32,46,48,45)(34,50,49,35)(41,52,55,54)(47,51,56,53);(1,3,4,8)(2,6,5,7)(10,15)(11,18)(16,20)(19,24)(27,36)(31,44)(35,50)(37,39)(38,42)(45,46)(51,53)(52,54);(11,18)(14,17)(19,24)(22,23)(25,28)(26,32)(27,37)(29,41)(30,40)(31,45)(33,48)(34,47)(35,51)(36,39)(38,52)(42,54)(43,55)(44,46)(49,56)(50,53);(9,12,13,21)(10,16,15,20)(11,19,18,24)(14,22,17,23)(26,33)(28,40)(31,44)(34,49)(35,50)(37,39)(41,55)(52,54)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=ghi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6406514);