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 | |
4263 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,9,10)(4,8,12,13)(11,14,15,16)(17,18,23,20)(19,21,24,27)(22,25,31,28)(26,29,30,32)(33,34)(35,42)(36,39)(37,40)(38,44)(41,46)(43,47)(45,48);(2,6)(4,12)(7,10)(11,15)(17,19,23,24)(18,21,20,27)(22,30,31,26)(25,32,28,29)(33,35,39,37)(34,40,36,42)(38,45,46,43)(41,48,44,47);(3,9)(7,10)(11,15)(14,16)(19,24)(21,27)(26,30)(29,32)(33,34,39,36)(35,40,37,42)(38,41,46,44)(43,47,45,48);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,20,23,18)(19,21,24,27)(22,28,31,25)(26,29,30,32)(33,36,39,34)(35,40,37,42)(38,44,46,41)(43,47,45,48);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,15)(13,16)(17,21)(18,24)(19,20)(22,29)(23,27)(25,30)(26,28)(31,32)(33,37,39,35)(34,40,36,42)(38,45,46,43)(41,47,44,48);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,22)(18,25)(19,26)(20,28)(21,29)(23,31)(24,30)(27,32)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=h,d^2=hi,e^2=h,f^2=gh,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, 10268138);