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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3706 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,7)(3,9,11,6)(4,8,13,15)(10,16,12,14)(17,18,22,26)(19,23,27,32)(20,24,30,29)(21,25,31,28)(33,34,38,39)(35,40,42,37)(36,44,46,41)(43,45,48,47);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(19,27)(21,31)(23,32)(25,28)(34,39)(35,42)(41,44)(43,48);(2,7)(4,13)(6,9)(10,12)(17,19)(18,23)(20,25)(21,29)(22,27)(24,31)(26,32)(28,30)(33,35)(34,40)(36,43)(37,39)(38,42)(41,47)(44,45)(46,48);(1,2)(3,6)(4,8)(5,7)(9,11)(10,14)(12,16)(13,15)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,20)(18,24)(19,28)(21,23)(22,30)(25,27)(26,29)(31,32);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,13)(15,16)(17,21)(18,25)(19,29)(20,23)(22,31)(24,27)(26,28)(30,32)(33,37,38,40)(34,35,39,42)(36,45,46,47)(41,43,44,48)Polycyclic presentation:
2^9|a^2=hi,b^2=h,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9148162);