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 | #2678 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4013 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,10)(4,5)(6,7)(8,11)(9,14)(12,15)(13,16)(17,18,22,25)(19,20,27,23)(21,24,29,31)(26,28,32,30)(33,34,39,40)(35,44,45,41)(36,48,42,37)(38,43,46,47);(2,7)(3,11)(4,14)(13,15)(17,19,22,27)(18,23,25,20)(21,26,29,32)(24,30,31,28)(34,40)(35,45)(36,42)(43,47);(2,7)(5,9)(8,10)(13,15)(21,29)(24,31)(26,32)(28,30)(33,35)(34,41)(36,46)(37,47)(38,42)(39,45)(40,44)(43,48);(1,3,6,11)(2,8,7,10)(4,12,14,16)(5,13,9,15)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,30)(31,32)(33,36,39,42)(34,37,40,48)(35,46,45,38)(41,47,44,43);(1,4)(2,9)(3,12)(5,7)(6,14)(8,15)(10,13)(11,16)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31)(33,37)(34,42)(35,47)(36,40)(38,41)(39,48)(43,45)(44,46);(1,5,6,9)(2,4,7,14)(3,13,11,15)(8,12,10,16)(17,21,22,29)(18,24,25,31)(19,26,27,32)(20,28,23,30)(33,38,39,46)(34,43,40,47)(35,42,45,36)(37,41,48,44)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8109460);