Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6660434 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23923 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1753 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,5)(4,6)(7,8)(10,13)(11,15)(17,18)(19,29)(20,34)(21,28)(22,26)(23,27)(24,31)(25,36)(30,44)(32,39)(33,40)(35,42)(37,41)(38,47)(43,45)(46,48);(3,7)(5,8)(9,10,12,13)(11,14,15,16)(17,19)(18,24)(20,30)(21,32)(22,31)(23,33)(25,38)(26,29)(27,39)(28,40)(34,48)(35,46)(36,45)(37,47)(41,43)(42,44);(1,3)(2,5)(4,7)(6,8)(10,13)(14,16)(17,20)(18,25)(19,30)(21,36)(22,35)(23,37)(24,38)(26,41)(27,34)(28,42)(29,43)(31,46)(32,45)(33,47)(39,48)(40,44);(3,7)(5,8)(9,11,12,15)(10,14,13,16)(18,26)(19,31)(20,35)(27,28)(32,33)(36,37)(38,43)(44,48);(2,6)(5,8)(17,21,22,23)(18,27,26,28)(19,32,31,33)(20,36,35,37)(24,39,29,40)(25,34,41,42)(30,45,46,47)(38,48,43,44)Polycyclic presentation:
2^9|b^2=fi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=di,d^c=dfh,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6568125);