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 | #40982 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4016 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,4)(5,13)(6,7)(8,14)(9,10)(11,16)(12,15)(17,18,23,24)(19,21,20,25)(22,30,31,26)(27,32,28,29)(33,34,38,41)(35,39,45,48)(36,40,46,37)(42,43,44,47);(1,3)(2,4)(5,11)(6,9)(7,10)(8,12)(13,16)(14,15)(18,24)(21,25)(26,30)(29,32)(33,35)(34,39)(36,44)(37,43)(38,45)(40,47)(41,48)(42,46);(2,7)(3,9)(5,14)(12,16)(17,19,23,20)(18,25,24,21)(22,28,31,27)(26,29,30,32)(33,36,38,46)(34,40,41,37)(35,42,45,44)(39,43,48,47);(1,4)(2,3)(5,12)(6,10)(7,9)(8,11)(13,15)(14,16)(17,20,23,19)(18,21,24,25)(22,28,31,27)(26,29,30,32)(33,37)(34,36)(35,43)(38,40)(39,44)(41,46)(42,48)(45,47);(1,2)(3,10)(4,9)(5,8)(6,7)(11,16)(12,15)(13,14)(17,21)(18,20)(19,24)(22,29)(23,25)(26,28)(27,30)(31,32)(33,36)(34,40)(35,44)(37,41)(38,46)(39,47)(42,45)(43,48);(1,5,6,14)(2,8,7,13)(3,11,9,15)(4,12,10,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,30)(25,32)(33,35)(34,39)(36,44)(37,43)(38,45)(40,47)(41,48)(42,46)Polycyclic presentation:
2^9|a^2=hi,c^2=hi,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8851621);