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 | |
3806 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,4,5)(6,9,14,16)(11,15,12,13)(17,18,23,20)(19,24,21,26)(22,25,30,28)(27,31,29,32)(33,34,39,36)(35,42,40,37)(38,44,46,41)(43,45,47,48);(1,3,7,4)(2,5,10,8)(6,11,14,12)(9,13,16,15)(17,19)(18,24)(20,26)(21,23)(22,27)(25,31)(28,32)(29,30)(33,34,39,36)(35,42,40,37)(38,41,46,44)(43,48,47,45);(17,18,23,20)(19,26,21,24)(22,25,30,28)(27,32,29,31)(33,35)(34,37)(36,42)(38,43)(39,40)(41,45)(44,48)(46,47);(1,4,7,3)(2,8,10,5)(6,11,14,12)(9,13,16,15)(17,20,23,18)(19,26,21,24)(22,25,30,28)(27,31,29,32)(33,36,39,34)(35,42,40,37)(38,41,46,44)(43,45,47,48);(1,5,7,8)(2,4,10,3)(6,13,14,15)(9,12,16,11)(17,21,23,19)(18,24,20,26)(22,29,30,27)(25,31,28,32)(33,37,39,42)(34,40,36,35)(38,45,46,48)(41,47,44,43);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,38,39,46)(34,41,36,44)(35,43,40,47)(37,45,42,48)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=hi,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10352197);