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 | |
4162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,3)(7,9)(8,10)(14,15)(17,18)(19,27)(20,29)(21,23)(22,26)(24,28)(25,31)(30,32)(33,34,39,38)(35,40,45,44)(36,46,47,41)(37,42,43,48);(1,2,6,3)(4,7,11,9)(5,8,13,10)(12,14,16,15)(17,19,23,28)(18,24,21,27)(20,30,31,22)(25,32,29,26)(33,35,39,45)(34,40,38,44)(36,37,47,43)(41,48,46,42);(4,11)(7,9)(12,16)(14,15)(18,21)(24,27)(25,29)(26,32)(33,36,39,47)(34,41,38,46)(35,43,45,37)(40,42,44,48);(1,3,6,2)(4,9,11,7)(5,8,13,10)(12,14,16,15)(17,20,23,31)(18,25,21,29)(19,22,28,30)(24,26,27,32)(33,37)(34,42)(35,36)(38,48)(39,43)(40,46)(41,44)(45,47);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,21,23,18)(19,27,28,24)(20,29,31,25)(22,32,30,26)(33,36)(34,41)(35,37)(38,46)(39,47)(40,48)(42,44)(43,45);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,22)(18,26)(19,20)(21,32)(23,30)(24,25)(27,29)(28,31)(33,38,39,34)(35,44,45,40)(36,46,47,41)(37,42,43,48)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=i,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8440351);