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 | |
4052 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(7,9)(8,13)(14,15)(17,18,23,24)(19,27,28,22)(20,29,32,25)(21,31,30,26);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19,23,28)(18,22,24,27)(20,30,32,21)(25,26,29,31)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,45)(46,48);(4,12)(8,13)(10,16)(14,15)(18,24)(22,27)(25,29)(26,31)(34,39)(37,47)(40,45)(41,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,20,23,32)(18,25,24,29)(19,21,28,30)(22,31,27,26)(33,35)(34,40)(36,42)(37,41)(38,44)(39,45)(43,46)(47,48);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(17,21)(18,26)(19,20)(22,29)(23,30)(24,31)(25,27)(28,32)(33,36)(34,41)(35,43)(37,45)(38,46)(39,48)(40,47)(42,44);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,22,23,27)(18,19,24,28)(20,31,32,26)(21,29,30,25)(33,37,38,47)(34,42,39,43)(35,41,44,48)(36,45,46,40)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=h,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8230351);