Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62307 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60702 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #2920 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1540 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,9,5,8,14,16)(3,10,12,13,11,7,15,4)(17,18)(19,28)(20,25)(21,26)(22,35)(23,27)(24,30)(29,43)(31,38)(32,40)(33,41)(34,42)(36,39)(37,46)(44,47)(45,48);(1,3)(2,7)(4,9)(5,11)(6,12)(8,10)(13,16)(14,15)(17,19)(18,24)(20,29)(21,30)(22,36)(23,31)(25,37)(26,38)(27,39)(28,35)(32,44)(33,45)(34,46)(40,43)(41,47)(42,48);(1,4,6,10,5,13,14,7)(2,3,9,12,8,11,16,15)(17,20)(18,25)(19,29)(21,34)(22,32)(23,33)(24,37)(26,42)(27,41)(28,43)(30,46)(31,45)(35,40)(36,44)(38,48)(39,47);(2,8)(4,13)(7,10)(9,16)(17,21,23,22)(18,26,27,35)(19,30,31,36)(20,32,33,34)(24,38,39,28)(25,40,41,42)(29,44,45,46)(37,43,47,48);(2,8)(3,11)(9,16)(12,15)(20,33)(25,41)(29,45)(32,34)(37,47)(40,42)(43,48)(44,46)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=h,f^2=h,g^2=i,b^a=bf,d^a=di,e^a=ei,f^a=fh,e^b=ei,f^b=fh,d^c=dhi,e^c=eh98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252945);