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 |
| prime radical | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3928 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,4)(5,13)(7,10)(11,15)(17,18)(19,23)(20,29)(21,31)(22,26)(24,30)(25,27)(28,32)(33,34,39,44)(35,43,38,40)(36,42,37,41)(45,47,46,48);(2,4)(3,9)(8,12)(11,15)(17,18,22,26)(19,23,28,32)(20,24,30,29)(21,25,27,31)(33,35,39,38)(34,40,44,43)(36,46,37,45)(41,48,42,47);(1,2,6,4)(3,7,9,10)(5,8,13,12)(11,14,15,16)(17,19,22,28)(18,23,26,32)(20,27,30,21)(24,31,29,25)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,47)(46,48);(1,3,6,9)(2,7,4,10)(5,11,13,15)(8,14,12,16)(17,20)(18,24)(19,27)(21,28)(22,30)(23,31)(25,32)(26,29)(33,36,39,37)(34,41,44,42)(35,45,38,46)(40,47,43,48);(1,4,6,2)(3,10,9,7)(5,12,13,8)(11,16,15,14)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,32)(29,30)(33,37,39,36)(34,42,44,41)(35,46,38,45)(40,48,43,47);(1,5,6,13)(2,8,4,12)(3,11,9,15)(7,14,10,16)(17,21,22,27)(18,25,26,31)(19,20,28,30)(23,24,32,29)(33,38,39,35)(34,43,44,40)(36,46,37,45)(41,48,42,47)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=gh,d^2=h,e^2=h,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, 8017796);