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 | |
4098 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,11)(7,9)(12,16)(14,15)(19,24)(21,26)(22,30)(25,28)(33,34,40,44)(35,42,37,41)(36,38,39,43)(45,48,47,46);(1,2,6,3)(4,7,11,9)(5,10,13,8)(12,15,16,14)(17,18,23,20)(19,26,24,21)(22,28,30,25)(27,29,31,32)(33,35,40,37)(34,41,44,42)(36,45,39,47)(38,46,43,48);(2,3)(7,9)(8,10)(14,15)(17,19,23,24)(18,21,20,26)(22,27,30,31)(25,29,28,32)(33,36,40,39)(34,38,44,43)(35,45,37,47)(41,46,42,48);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,20,23,18)(19,21,24,26)(22,28,30,25)(27,29,31,32)(33,37,40,35)(34,42,44,41)(36,45,39,47)(38,46,43,48);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,21)(18,24)(19,20)(22,29)(23,26)(25,31)(27,28)(30,32)(33,38,40,43)(34,36,44,39)(35,46,37,48)(41,45,42,47);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,39,40,36)(34,43,44,38)(35,45,37,47)(41,46,42,48)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=gh,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8748812);