Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62253 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | #62793 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2862 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8,6,10,4,9)(3,11,13,7,14,15,12,16)(17,18)(19,28)(20,32)(21,26)(22,36)(23,27)(24,30)(25,35)(29,44)(31,39)(33,42)(34,41)(37,40)(38,45)(43,47)(46,48);(1,3)(2,7)(4,12)(5,13)(6,14)(8,15)(9,11)(10,16)(17,19)(18,24)(20,29)(21,30)(22,37)(23,31)(25,38)(26,39)(27,40)(28,36)(32,48)(33,45)(34,46)(35,47)(41,43)(42,44);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,16,11)(17,20,23,35)(18,25,27,32)(19,29,31,47)(21,34,22,33)(24,38,40,48)(26,42,36,41)(28,43,39,44)(30,46,37,45);(1,4,6,5)(2,9,10,8)(3,12,14,13)(7,11,16,15)(17,21,23,22)(18,26,27,36)(19,30,31,37)(20,33,35,34)(24,39,40,28)(25,41,32,42)(29,45,47,46)(38,43,48,44);(1,4,6,5)(2,8,10,9)(3,13,14,12)(7,11,16,15)(17,21,23,22)(18,26,27,36)(19,30,31,37)(20,34,35,33)(24,39,40,28)(25,42,32,41)(29,46,47,45)(38,44,48,43)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=ch,e^a=ei,f^a=fh,e^b=ei,f^b=fh,d^c=dhi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250441);