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)(28,33,43,50)(29,34,44,51)(30,45,42,35)(32,46,49,40)(38,48,54,56)(39,53,55,47);(9,11,14,21)(10,15,18,24)(12,20,13,19)(16,23,17,22)(25,27)(26,32)(28,38)(29,39)(30,46)(31,41)(33,47)(34,48)(35,52)(36,49)(37,45)(40,42)(43,54)(44,55)(50,53)(51,56);(2,5)(6,8)(9,12,14,13)(10,16,18,17)(11,19,21,20)(15,22,24,23)(25,28,31,43)(26,33,36,50)(27,38,41,54)(29,42,44,30)(32,47,49,53)(34,35,51,45)(37,48,52,56)(39,40,55,46);(9,13,14,12)(10,17,18,16)(11,20,21,19)(15,23,24,22)(25,29)(26,34)(27,39)(28,42)(30,43)(31,44)(32,48)(33,35)(36,51)(37,53)(38,40)(41,55)(45,50)(46,54)(47,52)(49,56);(1,3,4,7)(2,6,5,8)(10,18)(15,24)(16,17)(22,23)(28,43)(29,44)(33,50)(34,51)(38,54)(39,55)(47,53)(48,56)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=hi,d^2=h,e^2=g,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, 6397630);