Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438997 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1341 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,6)(5,8)(17,18,23,28)(19,29,32,25)(20,24,34,40)(21,26,36,43)(22,38,30,27)(31,42,46,41)(33,39,35,47)(37,48,45,44);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,13)(14,16)(17,19,30,34,23,32,22,20)(18,24,38,25,28,40,27,29)(21,31,45,35,36,46,37,33)(26,39,48,41,43,47,44,42);(1,3,5,4)(2,7,8,6)(10,13)(14,15)(17,20,22,32,23,34,30,19)(18,25,27,24,28,29,38,40)(21,35,37,31,36,33,45,46)(26,42,44,47,43,41,48,39);(9,11,12,16)(10,14,13,15)(17,21)(18,26)(19,31)(20,33)(22,37)(23,36)(24,39)(25,41)(27,44)(28,43)(29,42)(30,45)(32,46)(34,35)(38,48)(40,47);(1,4,5,3)(2,6,8,7)(19,32)(20,34)(21,36)(24,40)(25,29)(26,43)(37,45)(44,48)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=f,d^2=h,e^2=fgi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372597);