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 | #54951 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4020 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,11,12,8)(4,5,10,9)(13,14,16,15)(17,18)(19,24)(20,26)(21,30)(22,28)(23,25)(27,29)(31,32)(33,34,39,42)(35,37,40,45)(36,38,41,47)(43,44,46,48);(2,7)(3,12)(4,10)(14,15)(17,19)(18,24)(20,29)(21,31)(22,25)(23,28)(26,27)(30,32)(33,35,39,40)(34,37,42,45)(36,43,41,46)(38,44,47,48);(2,7)(4,10)(8,11)(13,16)(18,25)(19,28)(21,29)(26,32)(35,40)(37,45)(43,46)(44,48);(1,3)(2,8)(4,13)(5,15)(6,12)(7,11)(9,14)(10,16)(17,20)(18,26)(19,21)(22,27)(23,31)(24,30)(25,32)(28,29)(33,36)(34,38)(35,43)(37,44)(39,41)(40,46)(42,47)(45,48);(1,4,6,10)(2,9,7,5)(3,13,12,16)(8,14,11,15)(17,21)(18,27)(19,20)(22,26)(23,29)(24,32)(25,30)(28,31)(33,37,39,45)(34,40,42,35)(36,44,41,48)(38,46,47,43);(1,5,6,9)(2,10,7,4)(3,14,12,15)(8,13,11,16)(17,22)(18,28)(19,25)(20,30)(21,26)(23,24)(27,31)(29,32)(33,38)(34,41)(35,44)(36,42)(37,46)(39,47)(40,48)(43,45)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8528565);