Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #113356 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #390 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1438 |
| prime radical | #60942 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
810 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,5,10,7,3,6,9)(4,8,14,12,15,11,13,16)(17,18)(19,27,29,23)(20,31,33,24)(21,26)(22,25)(28,45)(30,43,35,39)(32,48)(34,44,36,41)(37,42)(38,46)(40,47);(1,3)(2,7)(4,12)(5,9)(6,10)(8,13)(11,14)(15,16)(17,19)(18,23)(20,32)(21,35)(22,29)(24,40)(25,27)(26,43)(28,34)(30,37)(31,48)(33,47)(36,46)(38,41)(39,42)(44,45);(1,4)(2,8)(3,11)(5,14)(6,13)(7,15)(9,16)(10,12)(17,20)(18,24)(19,28)(21,36)(22,33)(23,38)(25,31)(26,44)(27,45)(29,46)(30,47)(32,35)(34,37)(39,48)(40,43)(41,42)Polycyclic presentation:
2^9|a^2=g,d^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,d^b=di,e^b=ehi,f^b=fh,g^b=ghi,d^c=dh,f^c=fh,g^c=ghi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 20557);