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 | |
4079 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,15)(10,16)(13,14)(17,18)(19,26)(20,23)(21,28)(22,24)(25,30)(27,31)(29,32)(33,34,40,38)(35,44,46,41)(36,42,37,47)(39,48,45,43);(1,2)(3,7)(4,8)(5,9)(6,11)(10,14)(12,15)(13,16)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(33,35,40,46)(34,41,38,44)(36,45,37,39)(42,48,47,43);(1,3)(2,6)(4,10)(5,11)(7,9)(8,13)(12,16)(14,15)(17,18,23,20)(19,26,22,24)(21,25,30,28)(27,32,29,31)(33,36,40,37)(34,42,38,47)(35,45,46,39)(41,48,44,43);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,20,23,18)(19,24,22,26)(21,28,30,25)(27,31,29,32)(33,37,40,36)(34,42,38,47)(35,39,46,45)(41,48,44,43);(1,2)(3,6)(4,8)(5,9)(7,11)(10,13)(12,15)(14,16)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,38,40,34)(35,44,46,41)(36,47,37,42)(39,48,45,43);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,22,23,19)(18,24,20,26)(21,29,30,27)(25,31,28,32)(33,39)(34,43)(35,36)(37,46)(38,48)(40,45)(41,42)(44,47)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,d^2=g,e^2=gh,f^2=h,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, 8044236);