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)(7,10)(11,15)(14,16)(18,22)(19,25)(24,27)(26,31)(33,34,38,39)(35,42,45,40)(36,37,46,41)(43,47,48,44);(1,2)(3,10)(4,12)(5,6)(7,9)(8,13)(11,14)(15,16)(17,18,21,22)(19,23,25,29)(20,27,28,24)(26,32,31,30)(34,39)(37,41)(40,42)(44,47);(2,6)(7,10)(8,12)(14,16)(19,25)(23,29)(26,31)(30,32)(33,35,38,45)(34,40,39,42)(36,43,46,48)(37,44,41,47);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,19,21,25)(18,23,22,29)(20,26,28,31)(24,30,27,32)(33,35)(34,40)(36,43)(37,44)(38,45)(39,42)(41,47)(46,48);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,18)(19,23)(20,24)(21,22)(25,29)(26,30)(27,28)(31,32)(33,36)(34,41)(35,43)(37,39)(38,46)(40,47)(42,44)(45,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,20,21,28)(18,24,22,27)(19,26,25,31)(23,30,29,32)(33,37,38,41)(34,36,39,46)(35,44,45,47)(40,43,42,48)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=h,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8659471);