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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3848 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,4,10,8)(5,13,14,9)(11,15,16,12)(18,20)(23,26)(24,28)(30,32)(33,34,38,41)(35,39,43,37)(36,44,46,40)(42,45,48,47);(2,7)(3,10)(9,13)(11,16)(19,25)(23,26)(27,31)(30,32)(33,35,38,43)(34,37,41,39)(36,42,46,48)(40,45,44,47);(2,7)(3,10)(9,13)(11,16)(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32)(33,35)(34,39)(36,42)(37,41)(38,43)(40,47)(44,45)(46,48);(1,3,6,10)(2,8,7,4)(5,11,14,16)(9,15,13,12)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,36,38,46)(34,40,41,44)(35,42,43,48)(37,45,39,47);(1,4,6,8)(2,3,7,10)(5,12,14,15)(9,11,13,16)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)(33,37,38,39)(34,35,41,43)(36,45,46,47)(40,42,44,48)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=ghi,d^2=g,e^2=h,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, 8066340);