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 | #57997 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4000 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,15)(10,16)(13,14)(17,18,23,27)(19,28,31,24)(20,21,25,32)(22,30,29,26)(33,34,38,39)(35,42,37,40)(36,44,46,41)(43,47,45,48);(1,2)(3,7)(4,8)(5,9)(6,11)(10,14)(12,15)(13,16)(17,19,23,31)(18,24,27,28)(20,29,25,22)(21,30,32,26)(34,39)(40,42)(41,44)(47,48);(1,3)(2,6)(4,10)(5,11)(7,9)(8,13)(12,16)(14,15)(17,20,23,25)(18,21,27,32)(19,29,31,22)(24,30,28,26)(33,35,38,37)(34,40,39,42)(36,43,46,45)(41,47,44,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,18)(19,24)(20,21)(22,26)(23,27)(25,32)(28,31)(29,30)(33,34)(35,40)(36,44)(37,42)(38,39)(41,46)(43,48)(45,47);(1,2)(3,6)(4,8)(5,9)(7,11)(10,13)(12,15)(14,16)(17,21)(18,25)(19,30)(20,27)(22,24)(23,32)(26,31)(28,29)(33,36,38,46)(34,41,39,44)(35,43,37,45)(40,47,42,48);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,22)(18,26)(19,20)(21,24)(23,29)(25,31)(27,30)(28,32)(33,37,38,35)(34,42,39,40)(36,45,46,43)(41,48,44,47)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8044276);