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 | |
4082 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(7,10)(8,12)(14,16)(17,18,23,20)(19,21,24,27)(22,25,31,28)(26,29,30,32)(33,34,40,44)(35,45,47,41)(36,39,37,42)(38,43,46,48);(1,2,5,6)(3,7,9,10)(4,12,13,8)(11,16,15,14)(17,19,23,24)(18,21,20,27)(22,30,31,26)(25,32,28,29)(33,35)(34,41)(36,43)(37,48)(38,39)(40,47)(42,46)(44,45);(3,9)(7,10)(11,15)(14,16)(19,24)(21,27)(26,30)(29,32)(33,36,40,37)(34,42,44,39)(35,43,47,48)(38,41,46,45);(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)(33,37,40,36)(34,39,44,42)(35,43,47,48)(38,41,46,45);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,21)(18,24)(19,20)(22,29)(23,27)(25,30)(26,28)(31,32)(33,38,40,46)(34,43,44,48)(35,39,47,42)(36,45,37,41);(1,4,5,13)(2,8,6,12)(3,11,9,15)(7,14,10,16)(17,22)(18,25)(19,26)(20,28)(21,29)(23,31)(24,30)(27,32)(33,39)(34,36)(35,46)(37,44)(38,47)(40,42)(41,48)(43,45)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,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, 10268139);