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 | #48187 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4044 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)(19,20)(21,24)(22,25)(23,27)(26,28)(29,30)(31,32)(33,34,38,42)(35,43,46,39)(36,47,45,40)(37,48,44,41);(1,3,7,4)(2,5,10,8)(6,11,14,12)(9,13,16,15)(19,23)(20,27)(26,31)(28,32)(33,34)(35,39)(36,40)(37,48)(38,42)(41,44)(43,46)(45,47);(17,19,22,23)(18,20,25,27)(21,26,30,31)(24,28,29,32)(33,35)(34,39)(36,44)(37,45)(38,46)(40,41)(42,43)(47,48);(1,4,7,3)(2,8,10,5)(6,11,14,12)(9,13,16,15)(17,18)(19,20)(21,29)(22,25)(23,27)(24,30)(26,32)(28,31)(33,36,38,45)(34,40,42,47)(35,44,46,37)(39,41,43,48);(1,5,7,8)(2,4,10,3)(6,13,14,15)(9,12,16,11)(17,20)(18,23)(19,25)(21,28)(22,27)(24,31)(26,29)(30,32)(33,37,38,44)(34,41,42,48)(35,45,46,36)(39,40,43,47);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,29)(27,32)(33,34)(35,39)(36,47)(37,41)(38,42)(40,45)(43,46)(44,48)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=i,d^2=g,e^2=g,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, 10330192);