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 | #37468 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3926 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,8)(3,6,9,12)(4,7,11,14)(10,13,15,16)(17,18)(19,25)(20,27)(21,29)(22,24)(23,30)(26,28)(31,32)(33,34,38,41)(35,39,43,37)(36,44,46,40)(42,45,48,47);(3,9)(6,12)(10,15)(13,16)(17,19)(18,20)(21,26)(22,27)(23,28)(24,25)(29,31)(30,32)(33,35,38,43)(34,37,41,39)(36,42,46,48)(40,45,44,47);(1,2)(3,6)(4,7)(5,8)(9,12)(10,13)(11,14)(15,16)(17,18,22,24)(19,20,27,25)(21,23,30,29)(26,28,32,31)(33,35)(34,39)(36,42)(37,41)(38,43)(40,47)(44,45)(46,48);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,20,22,25)(18,19,24,27)(21,28,30,31)(23,26,29,32)(33,36,38,46)(34,40,41,44)(35,42,43,48)(37,45,39,47);(1,2)(3,6)(4,7)(5,8)(9,12)(10,13)(11,14)(15,16)(17,19)(18,20)(21,26)(22,27)(23,28)(24,25)(29,31)(30,32);(1,4,5,11)(2,7,8,14)(3,10,9,15)(6,13,12,16)(17,21)(18,23)(19,26)(20,28)(22,30)(24,29)(25,31)(27,32)(33,37,38,39)(34,35,41,43)(36,45,46,47)(40,42,44,48)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=gi,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8065940);