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 | #45601 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4063 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18)(19,27)(20,29)(21,26)(22,30)(23,25)(24,28)(31,32)(33,34,38,43)(35,41,44,39)(36,45,46,40)(37,47,48,42);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,19,23,28)(18,24,25,27)(20,21,30,31)(22,26,29,32)(35,44)(36,46)(39,41)(40,45);(2,6)(7,10)(8,12)(14,16)(18,25)(19,28)(20,30)(26,32)(33,35,38,44)(34,39,43,41)(36,47,46,42)(37,40,48,45);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,20)(18,22)(19,21)(23,30)(24,26)(25,29)(27,32)(28,31)(33,36,38,46)(34,40,43,45)(35,42,44,47)(37,41,48,39);(1,2)(3,7)(4,12)(5,6)(8,13)(9,10)(11,16)(14,15)(17,21)(18,26)(19,20)(22,24)(23,31)(25,32)(27,29)(28,30)(33,35)(34,41)(36,42)(37,45)(38,44)(39,43)(40,48)(46,47);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,22)(18,20)(19,26)(21,24)(23,29)(25,30)(27,31)(28,32)(33,37,38,48)(34,42,43,47)(35,40,44,45)(36,41,46,39)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gh,d^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9930453);