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 | #48170 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4084 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)(19,27)(20,25)(21,22)(23,26)(24,31)(28,30)(29,32)(33,34,40,44)(35,37,38,41)(36,45,48,42)(39,47,46,43);(1,2)(3,7)(4,8)(5,9)(6,11)(10,14)(12,15)(13,16)(17,19)(18,24)(20,28)(21,29)(22,30)(23,31)(25,32)(26,27)(33,35,40,38)(34,41,44,37)(36,43,48,47)(39,45,46,42);(1,3)(2,6)(4,10)(5,11)(7,9)(8,13)(12,16)(14,15)(17,20,23,21)(18,25,26,22)(19,28,31,29)(24,32,27,30)(33,36)(34,42)(35,43)(37,46)(38,47)(39,41)(40,48)(44,45);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,21,23,20)(18,25,26,22)(19,29,31,28)(24,32,27,30)(33,37,40,41)(34,35,44,38)(36,46,48,39)(42,43,45,47);(1,2)(3,6)(4,8)(5,9)(7,11)(10,13)(12,15)(14,16)(17,22,23,25)(18,21,26,20)(19,30,31,32)(24,29,27,28)(33,38,40,35)(34,37,44,41)(36,47,48,43)(39,42,46,45);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,19)(18,24)(20,28)(21,29)(22,30)(23,31)(25,32)(26,27)(33,39,40,46)(34,43,44,47)(35,45,38,42)(36,41,48,37)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,d^2=g,e^2=g,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, 8044105);