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 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1328 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)(13,17)(14,18)(19,22)(20,23)(21,24)(25,26,31,36)(27,37,41,52)(28,42,44,33)(29,34,45,51)(30,46,43,35)(32,47,50,40)(38,54,55,49)(39,53,56,48);(9,11,14,21)(10,15,18,24)(12,20,13,19)(16,23,17,22)(25,27)(26,32)(28,38)(29,39)(30,47)(31,41)(33,48)(34,49)(35,52)(36,50)(37,46)(40,43)(42,53)(44,55)(45,56)(51,54);(2,5)(6,8)(9,12,14,13)(10,16,18,17)(11,19,21,20)(15,22,24,23)(25,28)(26,33)(27,38)(29,43)(30,45)(31,44)(32,48)(34,46)(35,51)(36,42)(37,49)(39,40)(41,55)(47,56)(50,53)(52,54);(9,13,14,12)(10,17,18,16)(11,20,21,19)(15,23,24,22)(25,29,31,45)(26,34,36,51)(27,39,41,56)(28,43,44,30)(32,49,50,54)(33,46,42,35)(37,53,52,48)(38,40,55,47);(1,3,4,7)(2,6,5,8)(10,18)(15,24)(16,17)(22,23)(28,44)(29,45)(33,42)(34,51)(38,55)(39,56)(48,53)(49,54)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,d^2=hi,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,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, 6397638);