Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667604 |
| 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 | #2619 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1644 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,13)(11,14,15,16)(17,18,23,22)(19,24,30,29)(20,25,33,37)(21,26,35,36)(27,38,43,31)(28,39,34,41)(32,40,46,48)(42,45,47,44);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(17,19)(18,24)(20,31)(21,34)(22,29)(23,30)(25,27)(26,41)(28,35)(32,45)(33,38)(36,39)(37,43)(40,47)(42,48)(44,46);(10,13)(14,16)(17,20,23,33)(18,25,22,37)(19,27,30,43)(21,32,35,46)(24,38,29,31)(26,40,36,48)(28,42,34,47)(39,45,41,44);(2,6)(5,7)(10,13)(14,16)(17,21)(18,26)(19,28)(20,32)(22,36)(23,35)(24,39)(25,40)(27,42)(29,41)(30,34)(31,44)(33,46)(37,48)(38,45)(43,47);(3,8)(5,7)(9,11,12,15)(10,14,13,16)(18,22)(19,29)(21,35)(24,30)(25,37)(27,31)(28,39)(32,46)(34,41)(38,43)(42,45)(44,47)Polycyclic presentation:
2^9|a^2=gi,b^2=fg,c^2=i,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6625583);