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 | |
3789 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18)(19,26)(20,23)(21,28)(22,24)(25,30)(27,31)(29,32)(33,34,39,37)(35,40,38,46)(36,43,45,41)(42,48,44,47);(1,2,7,6)(3,10,4,8)(5,13,14,9)(11,15,12,16)(17,19,23,22)(18,24,20,26)(21,29,30,27)(25,32,28,31)(33,35,39,38)(34,40,37,46)(36,42,45,44)(41,47,43,48);(1,3,7,4)(2,8,6,10)(5,11,14,12)(9,15,13,16)(17,18,23,20)(19,24,22,26)(21,25,30,28)(27,31,29,32)(33,36)(34,41)(35,42)(37,43)(38,44)(39,45)(40,47)(46,48);(1,4,7,3)(2,8,6,10)(5,12,14,11)(9,15,13,16)(17,20,23,18)(19,24,22,26)(21,28,30,25)(27,31,29,32)(33,37,39,34)(35,40,38,46)(36,43,45,41)(42,47,44,48);(1,5,7,14)(2,9,6,13)(3,11,4,12)(8,15,10,16)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,38,39,35)(34,40,37,46)(36,44,45,42)(41,47,43,48);(1,6,7,2)(3,8,4,10)(5,13,14,9)(11,15,12,16)(17,22,23,19)(18,24,20,26)(21,29,30,27)(25,31,28,32)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=gi,d^2=hi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8178278);