Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| 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 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1204 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,18)(12,21)(13,16)(14,17)(15,20)(19,23)(22,24)(25,26,31,33,29,35,30,36)(27,37,44,48,39,32,40,46)(28,41,45,50,42,34,43,49)(38,47,54,55,52,51,53,56);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,18)(21,24)(25,27,29,39)(26,32,35,37)(28,38,42,52)(30,40,31,44)(33,46,36,48)(34,47,41,51)(43,53,45,54)(49,55,50,56);(1,3,4,8)(2,5,6,7)(26,33)(30,31)(32,46)(34,49)(35,36)(37,48)(40,44)(41,50)(43,45)(47,55)(51,56)(53,54);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(25,28)(26,34)(27,38)(29,42)(30,43)(31,45)(32,47)(33,49)(35,41)(36,50)(37,51)(39,52)(40,53)(44,54)(46,55)(48,56);(2,6)(5,7)(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(27,39)(28,42)(32,37)(34,41)(40,44)(43,45)(46,48)(49,50)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=h,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6667835);