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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3848 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(3,10)(5,14)(11,13)(17,18,23,26)(19,20,21,24)(22,30,31,25)(27,32,29,28)(33,34,38,39)(35,37,42,40)(36,44,46,41)(43,47,48,45);(1,2,6,7)(3,8,10,15)(4,5,9,14)(11,16,13,12)(17,19,23,21)(18,24,26,20)(22,27,31,29)(25,32,30,28)(34,39)(35,42)(41,44)(43,48);(3,10)(8,15)(11,13)(12,16)(17,19,23,21)(18,24,26,20)(22,27,31,29)(25,32,30,28)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,3,6,10)(2,8,7,15)(4,13,9,11)(5,12,14,16)(17,20,23,24)(18,19,26,21)(22,28,31,32)(25,27,30,29)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,4,6,9)(2,5,7,14)(3,11,10,13)(8,16,15,12)(17,21,23,19)(18,20,26,24)(22,29,31,27)(25,28,30,32)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,5)(2,9)(3,12)(4,7)(6,14)(8,11)(10,16)(13,15)(17,22,23,31)(18,25,26,30)(19,27,21,29)(20,28,24,32)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=hi,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,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066139);