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 | #2673 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,4)(3,8,10,6)(5,9,15,12)(11,16,13,14)(17,18,23,21)(19,20,27,24)(22,25,30,29)(26,28,32,31)(33,34,38,42)(35,43,46,39)(36,47,45,40)(37,41,44,48);(1,3,7,10)(2,8,4,6)(5,13,15,11)(9,14,12,16)(17,19,23,27)(18,24,21,20)(22,26,30,32)(25,31,29,28)(33,35,38,46)(34,39,42,43)(36,37,45,44)(40,41,47,48);(3,10)(6,8)(11,13)(14,16)(17,18,23,21)(19,24,27,20)(22,29,30,25)(26,28,32,31)(33,36,38,45)(34,40,42,47)(35,44,46,37)(39,48,43,41);(1,4,7,2)(3,6,10,8)(5,9,15,12)(11,16,13,14)(17,20)(18,19)(21,27)(22,28)(23,24)(25,26)(29,32)(30,31)(33,35)(34,39)(36,44)(37,45)(38,46)(40,48)(41,47)(42,43);(1,5)(2,9)(3,11)(4,12)(6,14)(7,15)(8,16)(10,13)(17,21,23,18)(19,24,27,20)(22,29,30,25)(26,31,32,28)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,48)(45,47);(1,6)(2,3)(4,10)(5,14)(7,8)(9,11)(12,13)(15,16)(17,22)(18,25)(19,26)(20,28)(21,29)(23,30)(24,31)(27,32)(33,37,38,44)(34,41,42,48)(35,45,46,36)(39,47,43,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=g,d^2=hi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8268134);