Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7324436 |
| 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 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1284 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,13,16)(11,17,18,14)(12,20,21,15)(19,22,24,23)(25,26)(27,37)(28,34)(29,36)(30,32)(31,35)(33,41)(38,51)(39,45)(40,48)(42,46)(43,50)(44,49)(47,53)(52,56)(54,55);(1,2)(3,7)(4,6)(5,8)(9,11,13,18)(10,14,16,17)(12,19,21,24)(15,22,20,23)(26,32)(28,42)(33,45)(35,36)(37,48)(38,52)(43,44)(53,54);(1,3)(2,5)(4,8)(6,7)(25,27,29,39,30,40,31,41)(26,33,35,48,32,45,36,37)(28,38,44,54,42,52,43,53)(34,47,50,56,46,55,49,51);(2,6)(3,8)(11,18)(14,17)(19,24)(22,23)(25,28)(26,34)(27,38)(29,44)(30,42)(31,43)(32,46)(33,47)(35,50)(36,49)(37,51)(39,54)(40,52)(41,53)(45,55)(48,56);(2,6)(5,7)(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(25,29,30,31)(26,35,32,36)(27,39,40,41)(28,43,42,44)(33,48,45,37)(34,49,46,50)(38,53,52,54)(47,51,55,56)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=gi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7240396);