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 | |
4295 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,3)(7,9)(8,10)(14,15)(17,18)(19,26)(20,28)(21,25)(22,23)(24,30)(27,31)(29,32)(33,34,39,42)(35,40,44,38)(36,45,37,41)(43,48,46,47);(1,2,6,3)(4,7,11,9)(5,8,13,10)(12,14,16,15)(17,19,23,25)(18,21,22,26)(20,27,30,32)(24,31,28,29)(33,35)(34,40)(36,46)(37,43)(38,42)(39,44)(41,48)(45,47);(4,11)(5,13)(7,9)(8,10)(18,22)(21,26)(24,28)(29,31)(33,36,39,37)(34,41,42,45)(35,43,44,46)(38,48,40,47);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,20)(18,24)(19,27)(21,31)(22,28)(23,30)(25,32)(26,29)(33,37,39,36)(34,41,42,45)(35,43,44,46)(38,47,40,48);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,21,23,26)(18,25,22,19)(20,29,30,31)(24,27,28,32)(33,38)(34,35)(36,47)(37,48)(39,40)(41,46)(42,44)(43,45);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,22,23,18)(19,26,25,21)(20,28,30,24)(27,29,32,31)(33,35)(34,38)(36,46)(37,43)(39,44)(40,42)(41,47)(45,48)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=i,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8851061);