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 | |
4064 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18,22,24)(19,23,26,21)(20,25,28,31)(27,29,32,30)(34,39)(35,41)(40,47)(42,43);(1,2,6,7)(3,4,8,12)(5,9,13,14)(10,11,15,16)(17,19)(18,23)(20,27)(21,24)(22,26)(25,29)(28,32)(30,31)(33,34)(35,37)(36,40)(38,39)(41,46)(42,44)(43,48)(45,47);(2,7)(3,8)(5,13)(11,16)(18,24)(19,26)(20,28)(29,30)(35,41)(37,46)(42,43)(44,48);(1,3)(2,4)(5,10)(6,8)(7,12)(9,11)(13,15)(14,16)(17,20,22,28)(18,25,24,31)(19,27,26,32)(21,30,23,29)(33,35)(34,37)(36,43)(38,41)(39,46)(40,48)(42,45)(44,47);(1,4)(2,8)(3,7)(5,11)(6,12)(9,15)(10,14)(13,16)(17,21)(18,19)(20,29)(22,23)(24,26)(25,32)(27,31)(28,30)(33,36,38,45)(34,40,39,47)(35,42,41,43)(37,44,46,48);(1,5)(2,9)(3,10)(4,11)(6,13)(7,14)(8,15)(12,16)(17,19)(18,21)(20,27)(22,26)(23,24)(25,30)(28,32)(29,31)(33,37)(34,35)(36,44)(38,46)(39,41)(40,42)(43,47)(45,48)Polycyclic presentation:
2^9|a^2=h,b^2=hi,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8815726);