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 | #50099 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4046 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)(3,9)(4,8)(5,10)(6,7)(11,16)(12,13)(14,15)(17,18)(19,22)(20,27)(21,24)(23,28)(25,29)(26,32)(30,31)(33,34)(35,43)(36,41)(37,47)(38,39)(40,42)(44,48)(45,46);(1,3)(2,5)(4,12)(6,10)(7,9)(8,16)(11,14)(13,15)(19,25)(22,29)(26,31)(30,32)(33,35,40,38)(34,39,42,43)(36,44,37,45)(41,46,47,48);(2,7)(3,10)(8,15)(11,12)(17,18,21,24)(19,22,25,29)(20,23,28,27)(26,30,31,32)(33,36,40,37)(34,41,42,47)(35,44,38,45)(39,46,43,48);(1,3)(2,5)(4,11)(6,10)(7,9)(8,13)(12,14)(15,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,37,40,36)(34,41,42,47)(35,44,38,45)(39,48,43,46);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,16)(10,12)(17,19,21,25)(18,22,24,29)(20,26,28,31)(23,30,27,32)(33,38,40,35)(34,39,42,43)(36,45,37,44)(41,46,47,48);(1,5)(2,3)(4,13)(6,9)(7,10)(8,11)(12,15)(14,16)(17,20)(18,23)(19,26)(21,28)(22,30)(24,27)(25,31)(29,32)(33,39)(34,38)(35,42)(36,46)(37,48)(40,43)(41,45)(44,47)Polycyclic presentation:
2^9|b^2=gi,c^2=gh,d^2=gi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154130);