Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234485 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1337 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,31,36)(27,32,40,47)(28,34,42,50)(29,35,30,33)(37,45,52,56)(38,46,39,44)(41,48,43,49)(51,54,53,55);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(25,27,31,40)(26,32,36,47)(28,37,42,52)(29,38,30,39)(33,44,35,46)(34,45,50,56)(41,51,43,53)(48,54,49,55);(1,3)(2,5)(4,8)(6,7)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(26,33)(29,30)(32,44)(34,48)(35,36)(38,39)(41,43)(45,54)(46,47)(49,50)(51,53)(55,56);(3,8)(5,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28)(26,34)(27,37)(29,43)(30,41)(31,42)(32,45)(33,48)(35,49)(36,50)(38,53)(39,51)(40,52)(44,54)(46,55)(47,56);(2,6)(5,7)(10,18)(15,19)(16,17)(23,24)(25,29,31,30)(26,35,36,33)(27,38,40,39)(28,41,42,43)(32,46,47,44)(34,48,50,49)(37,51,52,53)(45,54,56,55)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=f,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7144540);