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 | |
4094 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)(3,4)(5,10)(6,7)(8,9)(11,12)(13,16)(14,15)(33,34,40,44)(35,45,48,41)(36,42,37,38)(39,43,47,46);(2,7)(3,9)(10,15)(11,13)(17,18,22,23)(19,24,29,32)(20,25,30,28)(21,26,31,27)(33,35)(34,41)(36,43)(37,46)(38,47)(39,42)(40,48)(44,45);(2,7)(4,8)(10,15)(12,16)(18,23)(20,30)(24,32)(26,27)(33,36,40,37)(34,42,44,38)(35,43,48,46)(39,45,47,41);(1,3,6,9)(2,8,7,4)(5,13,14,11)(10,12,15,16)(17,19)(18,24)(20,27)(21,25)(22,29)(23,32)(26,30)(28,31)(33,37,40,36)(34,42,44,38)(35,46,48,43)(39,45,47,41);(1,4,6,8)(2,9,7,3)(5,12,14,16)(10,13,15,11)(17,20)(18,25)(19,27)(21,24)(22,30)(23,28)(26,29)(31,32)(33,38,40,42)(34,36,44,37)(35,47,48,39)(41,43,45,46);(1,5,6,14)(2,10,7,15)(3,11,9,13)(4,12,8,16)(17,21)(18,26)(19,28)(20,24)(22,31)(23,27)(25,29)(30,32)(33,39,40,47)(34,43,44,46)(35,42,48,38)(36,45,37,41)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=gh,d^2=g,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^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8424786);