Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470003 |
| 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 | |
1320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,31,35)(27,36,40,50)(28,41,29,33)(30,44,46,34)(32,45,48,39)(37,52,38,51)(42,49,43,56)(47,53,55,54);(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,31,40)(26,32,35,48)(28,37,29,38)(30,45,46,39)(33,47,41,55)(34,50,44,36)(42,54,43,53)(49,52,56,51);(2,5)(6,8)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,28)(26,33)(27,37)(29,31)(30,42)(32,47)(34,49)(35,41)(36,51)(38,40)(39,53)(43,46)(44,56)(45,54)(48,55)(50,52);(11,18)(14,22)(17,19)(21,23)(25,29,31,28)(26,33,35,41)(27,38,40,37)(30,43,46,42)(32,47,48,55)(34,49,44,56)(36,51,50,52)(39,54,45,53);(1,3,4,7)(2,6,5,8)(10,16)(14,22)(15,24)(21,23)(28,29)(33,41)(37,38)(42,43)(47,55)(49,56)(51,52)(53,54)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,d^2=i,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6387921);