Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10104 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1438 |
| prime radical | #60951 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
837 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,5,9,6,7,4,12)(3,8,10,15,11,14,13,16)(17,18)(19,29)(20,34,38,43)(21,27)(22,26)(23,28)(24,31)(25,36,44,37)(30,39,46,48)(32,41)(33,40)(35,47,45,42);(2,7)(3,10)(4,5)(8,15)(11,13)(14,16)(17,19)(18,24)(20,35)(21,32)(22,31)(23,33)(25,42)(26,40)(27,29)(28,41)(30,37)(34,39)(36,46)(38,45)(43,48)(44,47);(1,3)(2,8)(4,13)(5,10)(6,11)(7,14)(9,15)(12,16)(17,20)(18,25)(19,30)(21,37)(22,36)(23,38)(24,39)(26,34)(27,43)(28,44)(29,42)(31,35)(32,45)(33,46)(40,47)(41,48)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=di,f^b=fh,g^b=ghi,d^c=dh,e^c=ei,f^c=fh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 20752);