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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4220 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)(4,6)(5,12)(7,11)(8,13)(10,14)(15,16)(20,27)(24,29)(25,31)(30,32)(33,34,38,39)(35,36,43,41)(37,45,46,40)(42,47,48,44);(1,3,6,11)(2,7,4,9)(5,10,13,16)(8,14,12,15)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)(33,34,38,39)(35,41,43,36)(37,40,46,45)(42,47,48,44);(2,4)(7,9)(8,12)(14,15)(18,22)(23,28)(24,29)(30,32)(34,39)(36,41)(40,45)(44,47);(1,3)(2,7)(4,9)(5,10)(6,11)(8,14)(12,15)(13,16)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,47)(46,48);(1,4,6,2)(3,9,11,7)(5,12,13,8)(10,15,16,14)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)(33,36)(34,35)(37,44)(38,41)(39,43)(40,42)(45,48)(46,47);(1,5,6,13)(2,8,4,12)(3,10,11,16)(7,14,9,15)(17,20)(18,24)(19,25)(21,27)(22,29)(23,30)(26,31)(28,32)(33,37,38,46)(34,40,39,45)(35,42,43,48)(36,44,41,47)Polycyclic presentation:
2^9|a^2=gh,b^2=h,d^2=ghi,e^2=g,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^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8025836);