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 | |
4233 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)(3,10)(4,7)(5,12)(6,8)(9,14)(11,15)(13,16)(19,26)(21,29)(23,24)(25,28)(34,37)(35,41)(36,44)(47,48);(1,3,7,6)(2,8,4,10)(5,11,14,13)(9,15,12,16)(17,18)(19,24)(20,22)(21,25)(23,26)(27,31)(28,29)(30,32)(33,34,38,37)(35,39,41,43)(36,45,44,40)(42,48,46,47);(1,2,7,4)(3,10,6,8)(5,9,14,12)(11,16,13,15)(17,19)(18,23)(20,24)(21,27)(22,26)(25,30)(28,31)(29,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,46)(45,48);(1,4,7,2)(3,8,6,10)(5,12,14,9)(11,15,13,16)(17,20,22,18)(19,24,26,23)(21,28,29,25)(27,31,32,30)(33,36,38,44)(34,40,37,45)(35,42,41,46)(39,47,43,48);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,19)(18,24)(20,23)(21,27)(22,26)(25,31)(28,30)(29,32)(33,35)(34,39)(36,46)(37,43)(38,41)(40,48)(42,44)(45,47);(1,6,7,3)(2,8,4,10)(5,13,14,11)(9,15,12,16)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,37,38,34)(35,43,41,39)(36,45,44,40)(42,48,46,47)Polycyclic presentation:
2^9|b^2=g,c^2=ghi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8044823);