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 | #42095 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4051 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,7,10,14)(4,5,8,13)(11,12,15,16)(17,18,23,27)(19,24,30,29)(20,31,21,25)(22,26,32,28)(33,34,39,42)(35,43,41,37)(36,46,38,40)(44,48,45,47);(1,3,6,10)(2,7,9,14)(4,11,8,15)(5,12,13,16)(17,19)(18,24)(20,26)(21,28)(22,25)(23,30)(27,29)(31,32)(33,35,39,41)(34,37,42,43)(36,44,38,45)(40,48,46,47);(3,10)(7,14)(11,15)(12,16)(17,20,23,21)(18,25,27,31)(19,28,30,26)(22,24,32,29)(33,36,39,38)(34,40,42,46)(35,44,41,45)(37,48,43,47);(1,4)(2,5)(3,11)(6,8)(7,12)(9,13)(10,15)(14,16)(17,21,23,20)(18,25,27,31)(19,26,30,28)(22,24,32,29)(33,34)(35,43)(36,40)(37,41)(38,46)(39,42)(44,47)(45,48);(1,3)(2,7)(4,11)(5,12)(6,10)(8,15)(9,14)(13,16)(17,18)(19,29)(20,31)(21,25)(22,26)(23,27)(24,30)(28,32)(33,37,39,43)(34,41,42,35)(36,47,38,48)(40,44,46,45);(1,5)(2,8)(3,12)(4,9)(6,13)(7,15)(10,16)(11,14)(17,22)(18,26)(19,25)(20,24)(21,29)(23,32)(27,28)(30,31)(33,38,39,36)(34,40,42,46)(35,45,41,44)(37,48,43,47)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=h,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8502931);