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 | |
4216 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)(3,11)(4,9)(5,12)(6,8)(7,14)(10,16)(13,15)(17,18,23,20)(19,21,24,27)(22,25,31,28)(26,29,30,32)(33,34,38,39)(35,36,42,43)(37,40,45,46)(41,44,47,48);(1,3,6,14)(2,7,8,11)(4,12,15,10)(5,9,16,13)(17,19,23,24)(18,21,20,27)(22,30,31,26)(25,32,28,29)(34,39)(36,43)(37,45)(41,47);(2,8)(4,15)(7,11)(10,12)(19,24)(21,27)(26,30)(29,32)(33,35,38,42)(34,36,39,43)(37,41,45,47)(40,44,46,48);(1,3)(2,7)(4,12)(5,13)(6,14)(8,11)(9,16)(10,15)(17,20,23,18)(19,21,24,27)(22,28,31,25)(26,29,30,32)(33,36,38,43)(34,35,39,42)(37,44,45,48)(40,41,46,47);(1,4,6,15)(2,9,8,13)(3,12,14,10)(5,7,16,11)(17,21)(18,24)(19,20)(22,29)(23,27)(25,30)(26,28)(31,32)(33,35)(34,36)(37,41)(38,42)(39,43)(40,44)(45,47)(46,48);(1,5,6,16)(2,10,8,12)(3,13,14,9)(4,7,15,11)(17,22)(18,25)(19,26)(20,28)(21,29)(23,31)(24,30)(27,32)(33,37)(34,40)(35,41)(36,44)(38,45)(39,46)(42,47)(43,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=ghi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10268118);