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 | #53004 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4072 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)(3,12)(4,14)(5,6)(7,9)(8,13)(10,16)(11,15)(17,18)(19,21)(20,25)(22,27)(23,24)(26,31)(28,30)(29,32)(33,34,39,42)(35,38,41,47)(36,37,40,46)(43,44,45,48);(1,3,7,13)(2,8,9,12)(4,15,16,5)(6,10,11,14)(17,19)(18,21)(20,22)(23,26)(24,31)(25,27)(28,29)(30,32)(33,35)(34,38)(36,43)(37,44)(39,41)(40,45)(42,47)(46,48);(2,9)(6,11)(8,12)(10,14)(18,24)(19,26)(22,28)(25,32)(35,41)(38,47)(43,45)(44,48);(1,4,7,16)(2,10,9,14)(3,5,13,15)(6,12,11,8)(17,20)(18,25)(19,28)(21,30)(22,26)(23,29)(24,32)(27,31)(33,36)(34,37)(35,43)(38,44)(39,40)(41,45)(42,46)(47,48);(1,5)(2,11)(3,4)(6,9)(7,15)(8,14)(10,12)(13,16)(17,21,23,31)(18,26,24,19)(20,27,29,30)(22,25,28,32)(33,37)(34,40)(35,44)(36,42)(38,45)(39,46)(41,48)(43,47);(1,6,7,11)(2,5,9,15)(3,10,13,14)(4,12,16,8)(17,22,23,28)(18,27,24,30)(19,20,26,29)(21,25,31,32)(33,38,39,47)(34,41,42,35)(36,44,40,48)(37,45,46,43)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8230916);