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) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(7,9)(8,13)(14,15)(33,34)(35,42)(36,44)(37,47)(38,39)(40,45)(41,46)(43,48);(1,2,5,6)(3,9,10,7)(4,8,12,13)(11,15,16,14)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,31)(30,32)(34,39)(36,46)(40,42)(43,47);(3,10)(4,12)(7,9)(8,13)(18,22)(20,29)(23,25)(26,30)(33,35,38,45)(34,40,39,42)(36,43,46,47)(37,41,48,44);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,19)(18,23)(20,30)(21,27)(22,25)(24,32)(26,29)(28,31)(33,36)(34,41)(35,43)(37,42)(38,46)(39,44)(40,48)(45,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,20)(18,24)(19,26)(21,29)(22,28)(23,31)(25,32)(27,30)(33,35)(34,42)(36,47)(37,44)(38,45)(39,40)(41,48)(43,46);(1,4,5,12)(2,8,6,13)(3,11,10,16)(7,14,9,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,29)(30,32)(33,37,38,48)(34,43,39,47)(35,44,45,41)(36,42,46,40)Polycyclic presentation:
2^9|b^2=gh,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,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, 8947552);