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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3990 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,9,11,7)(5,12,13,8)(10,14,16,15)(17,18)(19,22)(20,24)(21,23)(25,28)(26,29)(27,30)(31,32)(33,34,37,40)(35,38,42,46)(36,39,44,48)(41,45,43,47);(1,3)(2,7)(4,9)(5,10)(6,11)(8,14)(12,15)(13,16)(17,19)(18,22)(20,26)(21,25)(23,28)(24,29)(27,31)(30,32)(33,35)(34,38)(36,43)(37,42)(39,47)(40,46)(41,44)(45,48);(1,4,6,2)(3,9,11,7)(5,8,13,12)(10,14,16,15)(18,23)(22,28)(24,30)(29,32)(33,36)(34,39)(35,41)(37,44)(38,45)(40,48)(42,43)(46,47);(1,5,6,13)(2,8,4,12)(3,10,11,16)(7,14,9,15)(19,25)(22,28)(26,31)(29,32)(36,44)(39,48)(41,43)(45,47);(3,11)(7,9)(10,16)(14,15)(18,23)(22,28)(24,30)(29,32)(35,42)(38,46)(41,43)(45,47);(17,20,21,27)(18,24,23,30)(19,26,25,31)(22,29,28,32)Polycyclic presentation:
2^9|a^2=gi,c^2=g,d^2=g,f^2=h,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, 8007496);