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 | #37049 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4049 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,9,11,7)(4,12,13,8)(10,14,16,15)(17,18,23,21)(19,26,25,22)(20,28,29,24)(27,30,31,32)(33,34)(35,39)(36,44)(37,42)(38,41)(40,46)(43,48)(45,47);(2,6)(7,9)(8,12)(14,15)(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,32)(29,31)(33,35,38,37)(34,39,41,42)(36,43,46,45)(40,47,44,48);(2,6)(7,9)(8,12)(14,15)(20,29)(24,28)(27,31)(30,32)(33,34,38,41)(35,42,37,39)(36,40,46,44)(43,48,45,47);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,20)(18,24)(19,27)(21,28)(22,30)(23,29)(25,31)(26,32)(33,34)(35,42)(36,44)(37,39)(38,41)(40,46)(43,47)(45,48);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,21,23,18)(19,26,25,22)(20,24,29,28)(27,30,31,32)(33,36)(34,40)(35,43)(37,45)(38,46)(39,47)(41,44)(42,48);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,22,23,26)(18,25,21,19)(20,30,29,32)(24,31,28,27)(33,37,38,35)(34,39,41,42)(36,45,46,43)(40,47,44,48)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,e^2=g,f^2=gh,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, 8065894);