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 | #55012 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4091 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)(3,8)(4,10)(5,11)(6,16)(7,9)(12,15)(13,14)(17,18,22,25)(19,23,27,21)(20,24,30,32)(26,31,28,29)(33,34,37,41)(35,39,43,38)(36,40,45,44)(42,47,48,46);(1,3,7,6)(2,8,9,16)(4,12,14,11)(5,10,15,13)(17,19)(18,23)(20,26)(21,25)(22,27)(24,31)(28,30)(29,32)(33,35)(34,38)(36,42)(37,43)(39,41)(40,46)(44,47)(45,48);(2,9)(3,6)(4,14)(5,15)(17,19,22,27)(18,23,25,21)(20,28,30,26)(24,29,32,31)(33,34,37,41)(35,38,43,39)(36,44,45,40)(42,47,48,46);(1,4)(2,10)(3,12)(5,16)(6,11)(7,14)(8,15)(9,13)(17,19)(18,23)(20,28)(21,25)(22,27)(24,29)(26,30)(31,32)(33,34)(35,38)(36,40)(37,41)(39,43)(42,46)(44,45)(47,48);(1,5,7,15)(2,11,9,12)(3,13,6,10)(4,8,14,16)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,32)(27,28)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,46)(45,48);(1,6,7,3)(2,8,9,16)(4,11,14,12)(5,10,15,13)(17,21,22,23)(18,19,25,27)(20,29,30,31)(24,26,32,28)(33,36)(34,40)(35,42)(37,45)(38,46)(39,47)(41,44)(43,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=ghi,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8934485);