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 | #54784 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4072 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,11)(4,5)(6,10)(7,9)(8,14)(12,13)(15,16)(17,18)(19,22)(20,25)(21,26)(23,27)(24,29)(28,30)(31,32)(33,34)(35,38)(36,43)(37,40)(39,45)(41,48)(42,47)(44,46);(1,3,7,14)(2,8,9,11)(4,12,10,16)(5,15,6,13)(17,19,22,18)(20,25,27,23)(21,24,29,26)(28,30,32,31)(33,35)(34,38)(36,44)(37,42)(39,48)(40,47)(41,45)(43,46);(1,4,7,10)(2,5,9,6)(3,12,14,16)(8,15,11,13)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,36,37,45)(34,39,40,43)(35,41,42,44)(38,46,47,48);(2,9)(4,10)(8,11)(12,16)(18,19)(20,27)(24,26)(28,32)(35,42)(36,45)(38,47)(39,43);(1,5,7,6)(2,4,9,10)(3,13,14,15)(8,16,11,12)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(34,40)(35,42)(39,43)(41,44);(1,6,7,5)(2,10,9,4)(3,13,14,15)(8,16,11,12)Polycyclic presentation:
2^9|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, 8012753);