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 | |
4144 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(7,9)(8,13)(14,15)(19,25)(20,28)(23,26)(24,29)(33,34,37,41)(35,39,43,38)(36,44,46,40)(42,47,45,48);(1,2,5,6)(3,9,10,7)(4,8,12,13)(11,15,16,14)(17,18)(19,26)(20,29)(21,22)(23,25)(24,28)(27,31)(30,32)(33,35,37,43)(34,38,41,39)(36,42,46,45)(40,47,44,48);(3,10)(4,12)(7,9)(8,13)(18,22)(19,25)(24,29)(27,30)(33,35,37,43)(34,38,41,39)(36,45,46,42)(40,48,44,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,19)(18,23)(20,30)(21,25)(22,26)(24,32)(27,28)(29,31)(33,35)(34,39)(36,42)(37,43)(38,41)(40,48)(44,47)(45,46);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,20)(18,24)(19,27)(21,28)(22,29)(23,31)(25,30)(26,32)(33,34)(35,38)(36,40)(37,41)(39,43)(42,47)(44,46)(45,48);(1,4,5,12)(2,8,6,13)(3,11,10,16)(7,14,9,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,29)(30,32)(33,36,37,46)(34,40,41,44)(35,42,43,45)(38,47,39,48)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8840267);