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 | #52363 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4053 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(5,13)(8,10)(11,16)(14,15)(17,18,22,26)(19,25,21,23)(20,24,30,28)(27,32,29,31)(33,34,38,41)(35,42,39,36)(37,45,46,40)(43,44,47,48);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,19,22,21)(18,23,26,25)(20,27,30,29)(24,31,28,32)(33,35,38,39)(34,36,41,42)(37,43,46,47)(40,44,45,48);(2,3)(7,9)(8,10)(14,15)(17,18,22,26)(19,23,21,25)(20,24,30,28)(27,31,29,32)(33,36)(34,39)(35,41)(37,44)(38,42)(40,47)(43,45)(46,48);(1,3,6,2)(4,7,12,9)(5,10,13,8)(11,14,16,15)(17,18)(19,23)(20,28)(21,25)(22,26)(24,30)(27,32)(29,31)(33,35)(34,36)(37,43)(38,39)(40,44)(41,42)(45,48)(46,47);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,32)(26,28)(33,36)(34,39)(35,41)(37,44)(38,42)(40,47)(43,45)(46,48);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,21,22,19)(18,25,26,23)(20,29,30,27)(24,32,28,31)(33,37,38,46)(34,40,41,45)(35,43,39,47)(36,44,42,48)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=gh,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8074616);