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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3985 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)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,29)(21,25)(23,26)(24,28)(27,32)(30,31)(34,39)(37,45)(40,46)(44,48);(1,2,5,8)(3,6,9,10)(4,7,12,14)(11,13,15,16)(17,19)(18,22)(20,21)(23,27)(24,30)(25,29)(26,32)(28,31)(36,43)(40,46)(42,47)(44,48);(3,9)(4,12)(6,10)(7,14)(19,27)(20,28)(22,32)(24,29)(33,34,38,39)(35,37,41,45)(36,40,43,46)(42,44,47,48);(1,3)(2,6)(4,11)(5,9)(7,13)(8,10)(12,15)(14,16)(17,20)(18,24)(19,21)(22,30)(23,28)(25,32)(26,29)(27,31)(33,35,38,41)(34,37,39,45)(36,42,43,47)(40,44,46,48);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,14)(13,15)(17,21)(18,25)(19,28)(20,27)(22,24)(23,31)(26,30)(29,32)(33,36,38,43)(34,40,39,46)(35,42,41,47)(37,44,45,48);(1,4,5,12)(2,7,8,14)(3,11,9,15)(6,13,10,16)(17,22)(18,19)(20,30)(21,24)(23,32)(25,28)(26,27)(29,31)(33,37)(34,41)(35,39)(36,44)(38,45)(40,47)(42,46)(43,48)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8835222);