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 | |
4101 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,3)(7,9)(8,10)(14,15)(17,18)(19,22)(20,27)(21,24)(23,28)(25,29)(26,32)(30,31)(33,34,38,41)(35,37,44,47)(36,40,46,39)(42,48,43,45);(1,2,6,3)(4,7,11,9)(5,10,13,8)(12,15,16,14)(19,25)(22,29)(26,31)(30,32)(33,35,38,44)(34,37,41,47)(36,43,46,42)(39,48,40,45);(4,11)(7,9)(12,16)(14,15)(17,18,21,24)(19,22,25,29)(20,23,28,27)(26,30,31,32)(33,36,38,46)(34,39,41,40)(35,42,44,43)(37,45,47,48);(1,3,6,2)(4,9,11,7)(5,10,13,8)(12,15,16,14)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,36)(34,40)(35,43)(37,45)(38,46)(39,41)(42,44)(47,48);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,19)(18,22)(20,26)(21,25)(23,30)(24,29)(27,32)(28,31)(33,35)(34,37)(36,43)(38,44)(39,48)(40,45)(41,47)(42,46);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,20)(18,23)(19,26)(21,28)(22,30)(24,27)(25,31)(29,32)(33,37)(34,35)(36,45)(38,47)(39,42)(40,43)(41,44)(46,48)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9150404);