Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| 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 | #52492 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4083 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)(3,10)(4,8)(5,7)(6,9)(11,16)(12,15)(13,14)(17,18,21,23)(19,25,26,22)(20,24,28,30)(27,31,32,29)(33,34,37,40)(35,38,42,47)(36,43,45,39)(41,48,44,46);(1,3,6,5)(2,7,9,10)(4,11,12,13)(8,14,15,16)(17,19,21,26)(18,22,23,25)(20,27,28,32)(24,29,30,31)(33,35)(34,38)(36,44)(37,42)(39,48)(40,47)(41,45)(43,46);(1,4,6,12)(2,8,9,15)(3,11,5,13)(7,14,10,16)(18,23)(22,25)(24,30)(29,31)(33,36,37,45)(34,39,40,43)(35,41,42,44)(38,46,47,48);(2,9)(4,12)(7,10)(11,13)(18,23)(19,26)(24,30)(27,32)(35,42)(38,47)(41,44)(46,48);(1,3,6,5)(2,7,9,10)(4,11,12,13)(8,14,15,16)(17,20,21,28)(18,24,23,30)(19,27,26,32)(22,29,25,31);(1,5,6,3)(2,7,9,10)(4,13,12,11)(8,14,15,16)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gh,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,d^c=dghi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000294);