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 | |
4152 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,14)(10,12)(15,16)(17,18,23,21)(19,26,24,20)(22,29,31,25)(27,28,32,30)(33,34,39,43)(35,44,48,40)(36,41,46,38)(37,42,45,47);(1,2)(3,7)(4,11)(5,6)(8,13)(9,14)(10,16)(12,15)(17,19)(18,20)(21,26)(22,27)(23,24)(25,28)(29,30)(31,32)(33,35,39,48)(34,40,43,44)(36,45,46,37)(38,47,41,42);(2,6)(3,9)(8,11)(10,12)(17,20,23,26)(18,24,21,19)(22,30,31,28)(25,27,29,32)(33,36,39,46)(34,41,43,38)(35,45,48,37)(40,42,44,47);(1,3)(2,7)(4,12)(5,9)(6,14)(8,16)(10,13)(11,15)(17,20)(18,19)(21,24)(22,28)(23,26)(25,27)(29,32)(30,31)(33,36)(34,41)(35,45)(37,48)(38,43)(39,46)(40,42)(44,47);(1,4,5,13)(2,8,6,11)(3,10,9,12)(7,15,14,16)(17,21,23,18)(19,26,24,20)(22,29,31,25)(27,30,32,28)(33,37,39,45)(34,42,43,47)(35,46,48,36)(38,40,41,44);(1,2)(3,7)(4,8)(5,6)(9,14)(10,15)(11,13)(12,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,32)(26,30)(33,38,39,41)(34,36,43,46)(35,47,48,42)(37,40,45,44)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=g,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8838540);