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 | #46775 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3991 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)(3,6)(4,9)(5,8)(7,12)(10,13)(11,15)(14,16)(17,18,23,21)(19,25,20,24)(22,29,30,26)(27,31,28,32)(33,34)(35,43)(36,39)(37,47)(38,48)(40,46)(41,45)(42,44);(2,3)(7,9)(8,10)(14,15)(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32)(33,35)(34,40)(36,43)(37,44)(38,45)(39,46)(41,42)(47,48);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,19,23,20)(18,25,21,24)(22,27,30,28)(26,32,29,31)(33,34,39,36)(35,40,46,43)(37,47,45,41)(38,42,44,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,23,19)(18,25,21,24)(22,28,30,27)(26,32,29,31)(33,36,39,34)(35,40,46,43)(37,47,45,41)(38,48,44,42);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,21,23,18)(19,25,20,24)(22,29,30,26)(27,32,28,31)(33,37,39,45)(34,41,36,47)(35,44,46,38)(40,42,43,48);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,38,39,44)(34,42,36,48)(35,45,46,37)(40,41,43,47)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8027107);