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 | |
4056 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)(3,5)(4,11)(6,14)(7,9)(8,15)(10,12)(13,16)(17,18)(19,25)(20,27)(21,29)(22,24)(23,30)(26,28)(31,32)(33,34,39,44)(35,43,46,40)(36,47,48,41)(37,38,45,42);(1,3,7,11)(2,5,9,4)(6,10,15,16)(8,13,14,12)(17,18,22,24)(19,25,27,20)(21,23,30,29)(26,31,32,28)(33,35,39,46)(34,40,44,43)(36,45,48,37)(38,41,42,47);(1,4,7,5)(2,3,9,11)(6,13,15,12)(8,16,14,10)(17,19)(18,20)(21,26)(22,27)(23,28)(24,25)(29,31)(30,32)(33,36,39,48)(34,41,44,47)(35,45,46,37)(38,40,42,43);(1,5,7,4)(2,3,9,11)(6,13,15,12)(8,10,14,16)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31)(33,37)(34,42)(35,36)(38,44)(39,45)(40,47)(41,43)(46,48);(1,3)(2,5)(4,9)(6,10)(7,11)(8,13)(12,14)(15,16)(17,18)(19,20)(21,23)(22,24)(25,27)(26,28)(29,30)(31,32)(33,35)(34,43)(36,45)(37,48)(38,41)(39,46)(40,44)(42,47);(1,6)(2,8)(3,10)(4,12)(5,13)(7,15)(9,14)(11,16)(17,21,22,30)(18,23,24,29)(19,26,27,32)(20,28,25,31)(33,38,39,42)(34,37,44,45)(35,41,46,47)(36,43,48,40)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=gi,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8805766);