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 | #50349 | 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) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18)(19,20)(21,25)(22,28)(23,24)(26,27)(29,32)(30,31)(33,34,40,38)(35,39,45,41)(36,43,37,42)(44,46,47,48);(1,2)(3,9)(4,12)(5,6)(7,11)(8,13)(10,15)(14,16)(17,19,23,22)(18,20,24,28)(21,26,29,31)(25,27,32,30)(33,35,40,45)(34,41,38,39)(36,44,37,47)(42,46,43,48);(2,6)(4,13)(7,9)(10,14)(18,24)(19,22)(21,29)(27,30)(33,36,40,37)(34,42,38,43)(35,44,45,47)(39,48,41,46);(1,3)(2,7)(4,14)(5,11)(6,9)(8,16)(10,13)(12,15)(17,20)(18,19)(21,30)(22,24)(23,28)(25,31)(26,32)(27,29)(33,37,40,36)(34,43,38,42)(35,44,45,47)(39,48,41,46);(1,4)(2,8)(3,10)(5,13)(6,12)(7,15)(9,16)(11,14)(17,21)(18,25)(19,26)(20,27)(22,31)(23,29)(24,32)(28,30)(33,38,40,34)(35,39,45,41)(36,42,37,43)(44,46,47,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,13)(14,16)(17,22,23,19)(18,20,24,28)(21,31,29,26)(25,27,32,30)(33,39)(34,35)(36,46)(37,48)(38,45)(40,41)(42,47)(43,44)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8948097);