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 | #53164 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4116 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,4,9)(5,15,16,6)(11,12,13,14)(17,18)(19,20)(21,24)(22,30)(23,26)(25,32)(27,29)(28,31)(33,34)(35,41)(36,43)(37,45)(38,40)(39,47)(42,44)(46,48);(1,3,7,4)(2,8,10,9)(5,11,16,13)(6,14,15,12)(17,19,23,24)(18,20,26,21)(22,31,32,27)(25,29,30,28)(33,35,38,43)(34,36,40,41)(37,46,47,42)(39,48,45,44);(3,4)(8,9)(11,13)(12,14)(17,20,23,21)(18,24,26,19)(22,28,32,29)(25,31,30,27)(33,34,38,40)(35,36,43,41)(37,39,47,45)(42,44,46,48);(1,4,7,3)(2,9,10,8)(5,11,16,13)(6,14,15,12)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,31)(30,32)(33,36,38,41)(34,35,40,43)(37,44,47,48)(39,42,45,46);(1,5)(2,6)(3,11)(4,13)(7,16)(8,14)(9,12)(10,15)(17,21,23,20)(18,19,26,24)(22,29,32,28)(25,27,30,31)(33,35)(34,36)(37,42)(38,43)(39,44)(40,41)(45,48)(46,47);(1,6)(2,5)(3,12)(4,14)(7,15)(8,13)(9,11)(10,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,32)(24,31)(26,30)(33,37)(34,39)(35,42)(36,44)(38,47)(40,45)(41,48)(43,46)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=h,d^2=hi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8751776);