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 |
| prime radical | #36921 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4058 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)(3,6)(4,9)(5,10)(7,11)(8,13)(12,14)(15,16)(17,18)(19,26)(20,23)(21,28)(22,24)(25,30)(27,31)(29,32)(33,34,39,37)(35,41,45,40)(36,46,42,38)(43,44,47,48);(2,3)(7,9)(8,10)(14,15)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(33,35)(34,40)(36,43)(37,41)(38,44)(39,45)(42,47)(46,48);(2,3)(4,11)(8,10)(12,16)(17,18,23,20)(19,26,22,24)(21,25,30,28)(27,32,29,31)(33,35)(34,41)(36,47)(37,40)(38,44)(39,45)(42,43)(46,48);(1,3,6,2)(4,9,11,7)(5,10,13,8)(12,15,16,14)(17,20,23,18)(19,24,22,26)(21,28,30,25)(27,31,29,32)(33,36)(34,38)(35,43)(37,46)(39,42)(40,44)(41,48)(45,47);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,37,39,34)(35,41,45,40)(36,46,42,38)(43,48,47,44);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,22,23,19)(18,24,20,26)(21,29,30,27)(25,31,28,32)(33,38,39,46)(34,42,37,36)(35,44,45,48)(40,47,41,43)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=i,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8034693);