Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×C42 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C82 | C23 |
| socle | C22 | C2×#322 |
| prime radical | C23 | #322 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
196 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,3,7,4,9,11,15,5,8,10,16,12,6,13,14)(17,18)(19,23)(20,25)(21,26)(22,24)(27,34)(28,35)(29,33)(30,38)(31,36)(32,37)(39,45)(40,43)(41,44)(42,46)(47,48);(2,6,8,9)(3,10)(7,15,16,14)(11,13)(17,19,20,27,21,28,30,39,22,29,31,40,32,41,42,47)(18,23,25,34,26,35,38,45,24,33,36,43,37,44,46,48);(18,24)(23,33)(25,36)(26,37)(34,43)(35,44)(38,46)(45,48)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=f,e^2=g,f^2=h,g^2=i,b^a=bf,c^a=ci,e^a=eh,d^b=dh128 conjugacy classes — view the full character table.
| maximal subgroup of | #72231883, #72231889, #72231893, … |
| maximal normal subgroup of | #72231883, #72231889, #72231893, … |
| maximal quotient of | #72231884, #72231890, C2×#122215, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29411);