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 | #56341 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4065 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)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,23)(21,25)(22,26)(27,32)(28,31)(29,30)(33,34,39,43)(35,42,38,37)(36,41,47,40)(44,45,46,48);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19)(18,21)(20,27)(22,25)(23,30)(24,26)(28,29)(31,32)(33,35,39,38)(34,37,43,42)(36,44,47,46)(40,45,41,48);(3,10)(6,7)(11,16)(14,15)(20,28)(23,31)(27,29)(30,32)(33,36,39,47)(34,40,43,41)(35,44,38,46)(37,45,42,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20,22,28)(18,23,26,31)(19,27,25,29)(21,30,24,32)(33,36)(34,41)(35,44)(37,48)(38,46)(39,47)(40,43)(42,45);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,19)(18,24)(20,29)(21,26)(22,25)(23,30)(27,28)(31,32)(33,37)(34,35)(36,45)(38,43)(39,42)(40,44)(41,46)(47,48);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,21,22,24)(18,25,26,19)(20,30,28,32)(23,29,31,27)(33,38,39,35)(34,42,43,37)(36,46,47,44)(40,48,41,45)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9074629);