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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3908 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,5)(3,10,4,8)(6,13,14,9)(11,15,12,16)(17,18,22,26)(19,23,27,32)(20,24,30,29)(21,25,31,28)(33,34)(35,41)(36,40)(37,46)(38,42)(39,45)(43,47)(44,48);(3,4)(6,14)(8,10)(9,13)(19,27)(21,31)(23,32)(25,28)(33,35,38,45)(34,39,42,41)(36,43,46,48)(37,47,40,44);(1,3,7,4)(2,8,5,10)(6,11,14,12)(9,15,13,16)(17,19)(18,23)(20,25)(21,29)(22,27)(24,31)(26,32)(28,30)(33,36,38,46)(34,40,42,37)(35,43,45,48)(39,44,41,47);(1,4,7,3)(2,8,5,10)(6,12,14,11)(9,15,13,16)(17,19)(18,23)(20,25)(21,29)(22,27)(24,31)(26,32)(28,30)(33,37)(34,36)(35,44)(38,40)(39,48)(41,43)(42,46)(45,47);(1,5,7,2)(3,8,4,10)(6,13,14,9)(11,15,12,16)(17,20)(18,24)(19,28)(21,23)(22,30)(25,27)(26,29)(31,32)(33,36)(34,40)(35,43)(37,42)(38,46)(39,44)(41,47)(45,48);(1,6,7,14)(2,9,5,13)(3,11,4,12)(8,15,10,16)(17,21)(18,25)(19,29)(20,23)(22,31)(24,27)(26,28)(30,32)(33,35)(34,41)(36,43)(37,44)(38,45)(39,42)(40,47)(46,48)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=gh,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10235335);