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 | |
4239 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(4,12)(7,9)(10,16)(17,18)(19,22)(20,25)(21,24)(23,28)(26,29)(27,31)(30,32)(36,44)(40,47)(42,48)(45,46);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(18,19)(23,25)(24,26)(30,31)(33,34)(35,39)(36,40)(37,41)(38,43)(42,46)(44,47)(45,48);(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(33,35)(34,38)(36,42)(37,43)(39,41)(40,45)(44,48)(46,47);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,34)(35,38)(36,40)(37,41)(39,43)(42,45)(44,47)(46,48);(1,4,5,12)(2,8,6,13)(3,10,11,16)(7,14,9,15)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|c^2=h,d^2=h,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^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8042909);