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 | #49875 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4041 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,8)(3,4)(5,13)(15,16)(17,18,22,25)(19,21,24,31)(20,27,30,23)(26,32,28,29)(33,34,40,39)(35,43,38,41)(36,46,42,37)(44,45,47,48);(1,2)(3,9)(4,6)(5,14)(7,8)(10,13)(11,15)(12,16)(17,19,22,24)(18,21,25,31)(20,28,30,26)(23,32,27,29)(33,35,40,38)(34,41,39,43)(36,47,42,44)(37,48,46,45);(1,3,7,4)(2,9,8,6)(5,12,13,11)(10,15,14,16)(17,20,22,30)(18,23,25,27)(19,26,24,28)(21,29,31,32)(33,36)(34,37)(35,44)(38,47)(39,46)(40,42)(41,45)(43,48);(1,4,7,3)(2,9,8,6)(5,12,13,11)(10,16,14,15)(17,19)(18,21)(20,26)(22,24)(23,29)(25,31)(27,32)(28,30)(33,37,40,46)(34,42,39,36)(35,45,38,48)(41,47,43,44);(1,5)(2,10)(3,11)(4,12)(6,15)(7,13)(8,14)(9,16)(17,20)(18,23)(19,26)(21,29)(22,30)(24,28)(25,27)(31,32)(33,38,40,35)(34,43,39,41)(36,47,42,44)(37,48,46,45);(1,6,7,9)(2,4,8,3)(5,15,13,16)(10,12,14,11)(17,21)(18,24)(19,25)(20,29)(22,31)(23,28)(26,27)(30,32)(33,39,40,34)(35,43,38,41)(36,46,42,37)(44,48,47,45)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gi,d^2=g,e^2=ghi,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, 8538113);