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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3771 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)(3,9)(4,11)(5,13)(6,8)(7,14)(10,12)(15,16)(18,20)(19,22)(21,29)(31,32)(33,34,38,35)(36,41,44,39)(37,40,45,42)(43,47,48,46);(1,2,6,8)(3,9,11,4)(5,7,14,13)(10,15,16,12)(17,18,23,20)(19,24,22,27)(21,25,29,28)(26,31,30,32)(33,35,38,34)(36,41,44,39)(37,40,45,42)(43,46,48,47);(1,3)(2,4)(5,10)(6,11)(7,12)(8,9)(13,15)(14,16)(17,19,23,22)(18,24,20,27)(21,26,29,30)(25,31,28,32)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,16)(14,15)(17,20,23,18)(19,24,22,27)(21,28,29,25)(26,31,30,32);(1,2)(3,4)(5,7)(6,8)(9,11)(10,12)(13,14)(15,16)(17,21)(18,25)(19,26)(20,28)(22,30)(23,29)(24,31)(27,32)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47);(1,5,6,14)(2,7,8,13)(3,10,11,16)(4,12,9,15)(17,22,23,19)(18,24,20,27)(21,30,29,26)(25,31,28,32)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gi,d^2=gi,e^2=gh,f^2=g,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, 8662212);