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 | |
4143 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,11)(3,7,14,8)(4,9,10,5)(12,13,15,16)(17,18)(19,24)(20,23)(21,25)(22,26)(27,29)(28,31)(30,32)(33,34,38,42)(35,40,45,39)(36,46,41,37)(43,44,47,48);(1,3)(2,7)(4,15)(5,13)(6,14)(8,11)(9,16)(10,12)(17,19)(18,20)(21,27)(22,23)(24,26)(25,28)(29,32)(30,31)(33,35,38,45)(34,39,42,40)(36,43,41,47)(37,44,46,48);(1,3,6,14)(2,8,11,7)(4,15,10,12)(5,16,9,13)(17,20,22,24)(18,23,26,19)(21,29,30,28)(25,27,32,31)(33,35,38,45)(34,39,42,40)(36,47,41,43)(37,48,46,44);(1,3)(2,7)(4,15)(5,13)(6,14)(8,11)(9,16)(10,12)(17,20)(18,19)(21,28)(22,24)(23,26)(25,27)(29,30)(31,32)(33,36,38,41)(34,37,42,46)(35,43,45,47)(39,44,40,48);(1,4,6,10)(2,9,11,5)(3,12,14,15)(7,13,8,16)(17,19)(18,24)(20,26)(21,27)(22,23)(25,29)(28,32)(30,31)(33,35)(34,40)(36,47)(37,44)(38,45)(39,42)(41,43)(46,48);(1,5,6,9)(2,10,11,4)(3,13,14,16)(7,12,8,15)(17,21,22,30)(18,25,26,32)(19,27,23,31)(20,28,24,29)(33,37)(34,41)(35,44)(36,42)(38,46)(39,47)(40,43)(45,48)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=gi,d^2=hi,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8923664);