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 | |
4154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(5,13)(8,12)(10,16)(14,15)(17,18,23,27)(19,21,30,24)(20,25,32,29)(22,28,31,26)(33,34,39,42)(35,37,45,40)(36,46,41,38)(43,44,48,47);(2,4)(7,9)(8,12)(14,15)(17,19,23,30)(18,24,27,21)(20,28,32,26)(22,29,31,25)(33,35,39,45)(34,40,42,37)(36,43,41,48)(38,44,46,47);(1,2,6,4)(3,7,11,9)(5,8,13,12)(10,14,16,15)(17,20,23,32)(18,25,27,29)(19,28,30,26)(21,31,24,22)(33,36)(34,38)(35,43)(37,47)(39,41)(40,44)(42,46)(45,48);(1,3,6,11)(2,7,4,9)(5,10,13,16)(8,14,12,15)(17,21)(18,19)(20,31)(22,32)(23,24)(25,28)(26,29)(27,30)(33,37,39,40)(34,35,42,45)(36,47,41,44)(38,43,46,48);(1,4,6,2)(3,9,11,7)(5,12,13,8)(10,15,16,14)(17,19)(18,21)(20,28)(22,29)(23,30)(24,27)(25,31)(26,32)(33,35)(34,40)(36,43)(37,42)(38,44)(39,45)(41,48)(46,47);(1,5,6,13)(2,8,4,12)(3,10,11,16)(7,14,9,15)(17,22,23,31)(18,26,27,28)(19,29,30,25)(20,21,32,24)(33,38,39,46)(34,41,42,36)(35,44,45,47)(37,43,40,48)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=ghi,d^2=hi,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8017822);