Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #53837 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4163 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,4)(3,11,13,8)(5,9,6,10)(12,16,14,15)(17,18,21,23)(19,25,26,22)(20,28,30,24)(27,31,29,32)(33,34)(35,38)(36,43)(37,40)(39,45)(41,48)(42,47)(44,46);(1,3,7,13)(2,8,4,11)(5,14,6,12)(9,16,10,15)(17,19,21,26)(18,22,23,25)(20,29,30,27)(24,32,28,31)(33,35)(34,38)(36,44)(37,42)(39,48)(40,47)(41,45)(43,46);(1,4,7,2)(3,11,13,8)(5,10,6,9)(12,16,14,15)(17,18)(19,22)(20,24)(21,23)(25,26)(27,31)(28,30)(29,32)(33,36,37,45)(34,39,40,43)(35,41,42,44)(38,46,47,48);(2,4)(8,11)(9,10)(15,16)(18,23)(19,26)(24,28)(27,29)(35,42)(36,45)(38,47)(39,43);(1,5,7,6)(2,9,4,10)(3,12,13,14)(8,15,11,16)(17,20,21,30)(18,24,23,28)(19,27,26,29)(22,31,25,32)(34,40)(35,42)(39,43)(41,44);(1,6,7,5)(2,10,4,9)(3,12,13,14)(8,15,11,16)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012754);