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)(25,26,30,35)(27,37,40,32)(28,34,44,52)(29,36,31,33)(38,50,42,47)(39,46,41,48)(43,49,45,51)(53,56,54,55);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(25,27,30,40)(26,32,35,37)(28,42,44,38)(29,39,31,41)(33,46,36,48)(34,50,52,47)(43,54,45,53)(49,56,51,55);(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)(29,31)(32,46)(34,49)(35,36)(37,48)(39,41)(43,45)(47,55)(50,56)(51,52)(53,54);(3,8)(5,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28,30,44)(26,34,35,52)(27,38,40,42)(29,45,31,43)(32,47,37,50)(33,49,36,51)(39,54,41,53)(46,55,48,56);(2,6)(5,7)(10,18)(15,19)(16,17)(23,24)(25,29,30,31)(26,33,35,36)(27,39,40,41)(28,43,44,45)(32,46,37,48)(34,51,52,49)(38,53,42,54)(47,56,50,55)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fi,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, 7144542);