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 | |
4184 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,15)(11,16)(13,14)(17,18,23,26)(19,24,21,20)(22,25,31,30)(27,32,29,28)(33,34)(35,37)(36,44)(38,39)(40,42)(41,46)(43,47)(45,48);(1,2)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(17,19,23,21)(18,20,26,24)(22,27,31,29)(25,28,30,32)(34,39)(35,42)(41,44)(43,48);(1,3)(2,5)(4,11)(6,9)(7,10)(8,13)(12,16)(14,15)(17,20,23,24)(18,19,26,21)(22,28,31,32)(25,27,30,29)(33,34,38,39)(35,40,42,37)(36,41,46,44)(43,47,48,45);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,18)(19,20)(21,24)(22,30)(23,26)(25,31)(27,32)(28,29)(33,35)(34,40)(36,43)(37,39)(38,42)(41,47)(44,45)(46,48);(1,4,6,12)(2,8,10,15)(3,11,9,16)(5,13,7,14)(17,21,23,19)(18,24,26,20)(22,29,31,27)(25,32,30,28)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,5,6,7)(2,9,10,3)(4,13,12,14)(8,16,15,11)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,32)(26,30)(33,37,38,40)(34,35,39,42)(36,45,46,47)(41,43,44,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=g,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8366012);