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 | #41062 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3901 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)(3,5)(4,9)(6,16)(7,10)(8,14)(11,13)(12,15)(17,18,22,21)(19,23,28,26)(20,24,29,30)(25,31,27,32)(33,34)(35,44)(36,37)(38,46)(39,40)(41,45)(42,48)(43,47);(1,3,7,6)(2,5,10,16)(4,12,14,11)(8,13,9,15)(17,19)(18,23)(20,27)(21,26)(22,28)(24,32)(25,29)(30,31)(34,40)(35,45)(37,42)(43,46);(1,4,7,14)(2,8,10,9)(3,11,6,12)(5,15,16,13)(17,20,22,29)(18,24,21,30)(19,25,28,27)(23,31,26,32)(33,35)(34,41)(36,46)(37,47)(38,42)(39,45)(40,44)(43,48);(1,4)(2,9)(3,12)(5,15)(6,11)(7,14)(8,10)(13,16)(17,21,22,18)(19,26,28,23)(20,24,29,30)(25,31,27,32)(33,36)(34,42)(35,46)(37,40)(38,41)(39,48)(43,45)(44,47);(1,5)(2,3)(4,13)(6,10)(7,16)(8,12)(9,11)(14,15)(17,20)(18,24)(19,27)(21,30)(22,29)(23,32)(25,28)(26,31)(33,37,39,42)(34,36,40,48)(35,47,45,38)(41,46,44,43);(1,6,7,3)(2,5,10,16)(4,11,14,12)(8,13,9,15)(17,19)(18,23)(20,27)(21,26)(22,28)(24,32)(25,29)(30,31)(33,38,39,47)(34,43,40,46)(35,42,45,37)(36,41,48,44)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=gi,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=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, 8900918);