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 | |
4253 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,8,11)(3,12,7,9)(4,6,5,10)(13,15,14,16)(19,26)(21,29)(23,24)(25,28)(33,34,38,42)(35,41,37,39)(36,44,46,40)(43,47,45,48);(1,3,8,7)(2,9,11,12)(4,14,5,13)(6,15,10,16)(17,18)(19,24)(20,22)(21,25)(23,26)(27,31)(28,29)(30,32)(33,35,38,37)(34,39,42,41)(36,45,46,43)(40,48,44,47);(1,4,8,5)(2,10,11,6)(3,13,7,14)(9,15,12,16)(17,19)(18,23)(20,24)(21,27)(22,26)(25,30)(28,31)(29,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,46)(45,48);(1,5,8,4)(2,6,11,10)(3,14,7,13)(9,16,12,15)(17,20,22,18)(19,24,26,23)(21,28,29,25)(27,31,32,30)(33,36,38,46)(34,40,42,44)(35,43,37,45)(39,47,41,48);(1,6)(2,4)(3,15)(5,11)(7,16)(8,10)(9,14)(12,13)(17,19)(18,24)(20,23)(21,27)(22,26)(25,31)(28,30)(29,32)(33,34)(35,39)(36,44)(37,41)(38,42)(40,46)(43,48)(45,47);(1,7,8,3)(2,9,11,12)(4,14,5,13)(6,16,10,15)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,37,38,35)(34,41,42,39)(36,45,46,43)(40,48,44,47)Polycyclic presentation:
2^9|a^2=g,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, 8044824);