Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469956 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1362 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,21)(14,17)(18,22)(19,23)(20,24)(25,26)(27,35)(28,32)(29,42)(30,33)(31,38)(34,44)(36,51)(37,50)(39,53)(40,48)(41,56)(43,46)(45,52)(47,54)(49,55);(9,11)(10,15)(12,20)(13,18)(14,19)(16,24)(17,23)(21,22)(25,27,30,37)(26,31,33,46)(28,39,40,36)(29,43,44,38)(32,47,48,45)(34,50,42,35)(41,52,55,54)(49,51,56,53);(2,5)(6,8)(9,12,14,13)(10,16,17,21)(11,18,19,20)(15,22,23,24)(25,28,30,40)(26,32,33,48)(27,36,37,39)(29,41,44,55)(31,45,46,47)(34,49,42,56)(35,51,50,53)(38,52,43,54);(1,3,4,7)(2,6,5,8)(11,19)(15,23)(18,20)(22,24)(27,37)(28,40)(31,46)(32,48)(35,50)(38,43)(41,55)(49,56);(9,13,14,12)(10,16,17,21)(11,18,19,20)(15,24,23,22)(26,33)(31,46)(32,48)(34,42)(35,50)(45,47)(49,56)(51,53)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=g,e^2=hi,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=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6406744);