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 | #55059 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4047 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)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,18)(19,27)(20,21)(22,24)(23,25)(26,31)(28,29)(30,32)(33,34,39,42)(35,37,44,48)(36,41,38,40)(43,46,45,47);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19,23,22)(18,24,25,27)(20,30,26,28)(21,29,31,32)(33,35)(34,37)(36,45)(38,43)(39,44)(40,46)(41,47)(42,48);(3,10)(6,7)(11,16)(14,15)(18,25)(20,26)(24,27)(28,30)(33,36,39,38)(34,40,42,41)(35,43,44,45)(37,47,48,46);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20,23,26)(18,21,25,31)(19,28,22,30)(24,32,27,29)(33,37,39,48)(34,35,42,44)(36,46,38,47)(40,45,41,43);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,21)(18,26)(19,29)(20,25)(22,32)(23,31)(24,28)(27,30)(33,38,39,36)(34,41,42,40)(35,43,44,45)(37,47,48,46);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,22,23,19)(18,24,25,27)(20,28,26,30)(21,29,31,32)(33,35)(34,37)(36,43)(38,45)(39,44)(40,47)(41,46)(42,48)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gh,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8635730);