Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61591 |
| 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 | #62781 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2869 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,9,6,10,4,8)(3,11,12,7,13,15,14,16)(17,18)(19,28)(20,25)(21,26)(22,35)(23,27)(24,30)(29,42)(31,38)(32,40)(33,41)(34,45)(36,39)(37,44)(43,48)(46,47);(1,3)(2,7)(4,14)(5,12)(6,13)(8,11)(9,15)(10,16)(17,19)(18,24)(20,29)(21,30)(22,36)(23,31)(25,37)(26,38)(27,39)(28,35)(32,43)(33,44)(34,46)(40,47)(41,48)(42,45);(17,20)(18,25)(19,29)(21,33)(22,34)(23,32)(24,37)(26,41)(27,40)(28,42)(30,44)(31,43)(35,45)(36,46)(38,48)(39,47);(20,32)(25,40)(29,43)(33,34)(37,47)(41,45)(42,48)(44,46);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,16,11)(17,21,23,22)(18,26,27,35)(19,30,31,36)(20,33,32,34)(24,38,39,28)(25,41,40,45)(29,44,43,46)(37,48,47,42)Polycyclic presentation:
2^9|a^2=g,e^2=h,f^2=h,g^2=i,b^a=bf,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, 6250383);