Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1789 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)(3,6)(4,8)(5,7)(10,13)(11,15)(17,18)(19,27)(20,31)(21,23)(22,36)(24,30)(25,33)(26,35)(28,42)(29,44)(32,41)(34,40)(37,43)(38,48)(39,46)(45,47);(1,3,6,2)(4,8,5,7)(9,10,12,13)(11,14,15,16)(18,23)(19,28)(25,31)(26,36)(29,32)(30,43)(40,47)(46,48);(10,13)(14,16)(17,19,22,30,21,28,35,43)(18,24,26,27,23,37,36,42)(20,32,34,48,33,29,45,46)(25,39,40,44,31,38,47,41);(1,4,6,5)(2,7,3,8)(9,11,12,15)(10,14,13,16)(17,20)(18,25)(19,29)(21,33)(22,34)(23,31)(24,38)(26,40)(27,41)(28,32)(30,46)(35,45)(36,47)(37,39)(42,44)(43,48);(1,5,6,4)(2,7,3,8)(20,33)(25,31)(29,32)(34,45)(38,39)(40,47)(41,44)(46,48)Polycyclic presentation:
2^9|b^2=fi,c^2=g,d^2=fi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6705496);