Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #55100 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3976 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)(3,4)(5,13)(6,7)(8,9)(10,14)(11,15)(12,16)(20,27)(24,29)(25,31)(30,32)(34,37)(39,42)(40,45)(47,48);(2,7)(3,9)(10,13)(11,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)(33,34,38,37)(35,39,43,42)(36,40,46,45)(41,47,44,48);(2,7)(4,8)(10,13)(12,15)(18,22)(23,28)(24,29)(30,32)(33,35,38,43)(34,39,37,42)(36,44,46,41)(40,48,45,47);(1,3,6,9)(2,8,7,4)(5,11,14,16)(10,15,13,12)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,34,38,37)(35,39,43,42)(36,40,46,45)(41,47,44,48);(1,4,6,8)(2,9,7,3)(5,12,14,15)(10,16,13,11)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)(33,36,38,46)(34,40,37,45)(35,41,43,44)(39,47,42,48);(1,5,6,14)(2,10,7,13)(3,11,9,16)(4,12,8,15)(17,20)(18,24)(19,25)(21,27)(22,29)(23,30)(26,31)(28,32)(33,37,38,34)(35,42,43,39)(36,45,46,40)(41,48,44,47)Polycyclic presentation:
2^9|b^2=gh,c^2=gh,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8025159);