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 | |
4100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,9,11,4)(5,13,15,8)(10,12,14,16)(17,18)(19,22)(20,28)(21,24)(23,29)(25,27)(26,31)(30,32)(33,34)(35,41)(36,43)(37,45)(38,40)(39,46)(42,44)(47,48);(1,3,6,11)(2,4,7,9)(5,14,15,10)(8,16,13,12)(17,18)(19,25)(20,28)(21,24)(22,27)(23,29)(26,30)(31,32)(33,34,38,40)(35,41,43,36)(37,39,46,45)(42,47,48,44);(2,7)(4,9)(8,13)(12,16)(17,18,21,24)(19,25,27,22)(20,23,29,28)(26,31,32,30)(33,35)(34,36)(37,42)(38,43)(39,44)(40,41)(45,47)(46,48);(1,2)(3,4)(5,8)(6,7)(9,11)(10,12)(13,15)(14,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,29)(31,32)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,48)(46,47);(1,4)(2,3)(5,12)(6,9)(7,11)(8,10)(13,14)(15,16)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32)(33,34)(35,36)(37,39)(38,40)(41,43)(42,44)(45,46)(47,48);(1,5,6,15)(2,8,7,13)(3,10,11,14)(4,12,9,16)(17,20)(18,23)(19,26)(21,29)(22,30)(24,28)(25,31)(27,32)(33,37,38,46)(34,39,40,45)(35,42,43,48)(36,44,41,47)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=gh,f^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=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8669227);