Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #392 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1247 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)(12,16)(13,15)(14,18)(19,23)(20,22)(21,24)(25,26,29,34,30,33,40,46)(27,31,37,42,36,43,49,53)(28,32,39,45,38,44,50,54)(35,41,48,52,47,51,55,56);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,24)(22,23)(25,27)(26,31)(28,35)(29,37)(30,36)(32,41)(33,43)(34,42)(38,47)(39,48)(40,49)(44,51)(45,52)(46,53)(50,55)(54,56);(2,5)(6,7)(11,18)(14,17)(19,24)(21,23)(25,28)(26,32)(27,35)(29,39)(30,38)(31,41)(33,44)(34,45)(36,47)(37,48)(40,50)(42,52)(43,51)(46,54)(49,55)(53,56);(1,3,4,8)(2,6,5,7)(10,15)(11,18)(16,22)(19,24)(28,38)(32,44)(35,47)(39,50)(41,51)(45,54)(48,55)(52,56);(9,12,13,20)(10,16,15,22)(11,19,18,24)(14,21,17,23)(26,33)(27,36)(32,44)(34,46)(35,47)(37,49)(45,54)(48,55)Polycyclic presentation:
2^9|a^2=h,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=cfg,d^b=df,e^b=ei,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6314978);