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 | #58354 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3990 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(6,10)(11,15)(14,16)(17,18,22,26)(19,25,21,23)(20,24,29,32)(27,31,28,30)(33,34)(35,41)(36,44)(37,42)(38,39)(40,46)(43,45)(47,48);(1,2)(3,10)(4,12)(5,8)(6,9)(7,13)(11,14)(15,16)(19,21)(23,25)(27,28)(30,31)(34,39)(35,42)(40,44)(43,48);(1,2,5,8)(3,6,9,10)(4,7,13,12)(11,14,15,16)(17,19,22,21)(18,23,26,25)(20,28,29,27)(24,31,32,30)(33,35)(34,37)(36,43)(38,42)(39,41)(40,45)(44,47)(46,48);(1,2)(3,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,29)(18,24,26,32)(19,27,21,28)(23,30,25,31)(33,36,38,46)(34,40,39,44)(35,43,42,48)(37,45,41,47);(1,3,5,9)(2,6,8,10)(4,11,13,15)(7,14,12,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,34)(35,37)(36,40)(38,39)(41,42)(43,45)(44,46)(47,48);(1,4)(2,7)(3,11)(5,13)(6,14)(8,12)(9,15)(10,16)(17,21,22,19)(18,25,26,23)(20,28,29,27)(24,31,32,30)(33,37)(34,35)(36,45)(38,41)(39,42)(40,43)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=gh,c^2=gi,d^2=h,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8664405);