Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6660395 |
| 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 | |
1173 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(3,7)(6,8)(9,10)(11,18)(12,16)(13,15)(14,17)(19,23)(20,24)(21,22)(25,26,29,34,30,35,41,32)(27,31,38,45,37,44,51,42)(28,33,40,48,39,47,52,46)(36,43,49,54,50,55,56,53);(2,5)(6,8)(9,11,14,13)(10,15,17,18)(12,19,20,21)(16,22,24,23)(25,27)(26,31)(28,36)(29,38)(30,37)(32,42)(33,43)(34,45)(35,44)(39,50)(40,49)(41,51)(46,53)(47,55)(48,54)(52,56);(1,2)(3,6)(4,5)(7,8)(9,12,14,20)(10,16,17,24)(11,19,13,21)(15,22,18,23)(26,32)(29,41)(31,42)(33,46)(34,35)(38,51)(40,52)(43,53)(44,45)(47,48)(49,56)(54,55);(2,5)(6,8)(10,17)(12,20)(15,18)(19,21)(25,28,30,39)(26,33,35,47)(27,36,37,50)(29,40,41,52)(31,43,44,55)(32,46,34,48)(38,49,51,56)(42,53,45,54);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,18,17,15)(12,21,20,19)(16,23,24,22)(27,37)(28,39)(31,44)(33,47)(38,51)(40,52)(42,45)(46,48)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,d^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,e^b=ei,d^c=dfh,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6574647);