Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1213 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)(3,7)(4,8)(5,6)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,30,33)(27,37,40,32)(28,42,44,34)(29,35,31,36)(38,46,41,50)(39,47,53,51)(43,49,45,48)(52,56,54,55);(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27,30,40)(26,32,33,37)(28,39,44,53)(29,41,31,38)(34,47,42,51)(35,46,36,50)(43,52,45,54)(48,56,49,55);(2,6)(7,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(26,33)(27,38)(29,31)(32,46)(34,42)(37,50)(39,52)(40,41)(43,45)(47,55)(51,56)(53,54);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(25,28)(26,34)(27,39)(29,45)(30,44)(31,43)(32,47)(33,42)(35,49)(36,48)(37,51)(38,52)(40,53)(41,54)(46,55)(50,56);(1,4,5,3)(2,8,6,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,29)(26,35)(27,40)(28,43)(30,31)(32,37)(33,36)(34,48)(42,49)(44,45)(52,54)(55,56)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fi,d^2=fg,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6651604);