Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61569 |
| 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 | #62805 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,9,6,8,4,13)(3,10,12,7,11,16,14,15)(17,18)(19,27)(20,31)(21,34)(22,25)(23,29)(24,32)(26,36)(28,42)(30,40)(33,39)(35,38)(37,48)(41,43)(44,45)(46,47);(1,3,6,11)(2,7,8,15)(4,14,5,12)(9,16,13,10)(17,19,22,30)(18,23,25,38)(20,28,32,43)(21,35,36,29)(24,37,31,45)(26,40,34,27)(33,48,47,44)(39,42,46,41);(2,8)(7,15)(9,13)(10,16)(17,20)(18,24)(19,28)(21,33)(22,32)(23,37)(25,31)(26,39)(27,41)(29,44)(30,43)(34,46)(35,48)(36,47)(38,45)(40,42);(20,32)(24,31)(28,43)(33,47)(37,45)(39,46)(41,42)(44,48);(2,8)(3,11)(9,13)(12,14)(18,25)(23,38)(24,31)(26,34)(27,40)(37,45)(39,46)(41,42)Polycyclic presentation:
2^9|a^2=g,b^2=h,f^2=h,g^2=i,b^a=bf,c^a=ch,e^a=eh,f^a=fh,e^b=ei,f^b=fh,d^c=dhi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250396);