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 | |
4076 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,11,13,8)(4,9,14,12)(5,15,16,10)(17,18,22,25)(19,23,26,30)(20,21,24,29)(27,28,31,32)(33,34)(35,40)(36,45)(37,41)(38,46)(39,43)(42,47)(44,48);(2,7)(4,14)(8,11)(10,15)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(33,35,39,44)(34,40,43,48)(36,46,41,42)(37,47,45,38);(2,7)(5,16)(8,11)(9,12)(19,26)(23,30)(27,31)(28,32)(33,36,39,41)(34,37,43,45)(35,42,44,46)(38,48,47,40);(1,3,6,13)(2,8,7,11)(4,15,14,10)(5,12,16,9)(17,20)(18,21)(19,27)(22,24)(23,28)(25,29)(26,31)(30,32)(33,34)(35,40)(36,37)(38,42)(39,43)(41,45)(44,48)(46,47);(1,4)(2,9)(3,10)(5,11)(6,14)(7,12)(8,16)(13,15)(17,21)(18,24)(19,28)(20,25)(22,29)(23,31)(26,32)(27,30)(33,37)(34,41)(35,38)(36,43)(39,45)(40,46)(42,48)(44,47);(1,5,6,16)(2,10,7,15)(3,12,13,9)(4,11,14,8)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(33,38)(34,42)(35,37)(36,40)(39,47)(41,48)(43,46)(44,45)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=gh,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8800716);