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 | |
4214 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)(3,8)(4,6)(5,9)(7,11)(10,15)(12,13)(14,16)(17,18,23,22)(19,26,21,24)(20,28,30,25)(27,31,29,32)(33,34)(35,36)(37,45)(38,39)(40,47)(41,44)(42,46)(43,48);(1,3)(2,7)(4,8)(5,10)(6,11)(9,14)(12,15)(13,16)(17,19,23,21)(18,24,22,26)(20,29,30,27)(25,32,28,31)(33,35,38,41)(34,36,39,44)(37,42,47,48)(40,43,45,46);(5,13)(9,12)(10,16)(14,15)(18,22)(24,26)(25,28)(31,32)(34,39)(35,41)(40,45)(42,48);(1,4,6,2)(3,8,11,7)(5,12,13,9)(10,15,16,14)(17,20,23,30)(18,25,22,28)(19,27,21,29)(24,31,26,32)(33,36)(34,35)(37,46)(38,44)(39,41)(40,48)(42,45)(43,47);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,14)(13,16)(17,21,23,19)(18,26,22,24)(20,27,30,29)(25,31,28,32)(33,37,38,47)(34,40,39,45)(35,42,41,48)(36,43,44,46);(1,5)(2,9)(3,10)(4,12)(6,13)(7,14)(8,15)(11,16)(17,22,23,18)(19,26,21,24)(20,28,30,25)(27,32,29,31)(33,34)(35,36)(37,40)(38,39)(41,44)(42,43)(45,47)(46,48)Polycyclic presentation:
2^9|a^2=hi,b^2=h,d^2=g,e^2=h,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, 8231691);