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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,3)(7,9)(8,10)(14,15)(17,18)(19,24)(20,26)(21,31)(22,27)(23,25)(28,30)(29,32)(34,38)(35,41)(40,43)(42,47);(1,2,6,3)(4,7,11,9)(5,8,13,10)(12,14,16,15)(17,19)(18,24)(20,25)(21,28)(22,29)(23,26)(27,32)(30,31)(33,34,37,38)(35,39,41,45)(36,40,44,43)(42,46,47,48);(4,11)(7,9)(12,16)(14,15)(18,25)(19,26)(21,27)(29,30)(33,35,37,41)(34,39,38,45)(36,42,44,47)(40,46,43,48);(1,3,6,2)(4,9,11,7)(5,8,13,10)(12,14,16,15)(17,20,23,24)(18,26,25,19)(21,30,27,29)(22,32,31,28)(33,34)(35,39)(36,43)(37,38)(40,44)(41,45)(42,48)(46,47);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,21,23,27)(18,22,25,31)(19,28,26,32)(20,30,24,29)(33,35)(34,39)(36,42)(37,41)(38,45)(40,46)(43,48)(44,47);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,22)(18,27)(19,29)(20,28)(21,25)(23,31)(24,32)(26,30)(33,36,37,44)(34,40,38,43)(35,42,41,47)(39,46,45,48)Polycyclic presentation:
2^9|b^2=g,c^2=hi,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8441663);