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 | |
4238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,9)(3,7,10,6)(4,12,13,8)(11,14,16,15)(17,18,21,24)(19,22,26,29)(20,23,27,30)(25,28,31,32)(33,34)(35,44)(36,37)(38,41)(39,40)(42,43)(45,48)(46,47);(3,10)(6,7)(11,16)(14,15)(17,19,21,26)(18,22,24,29)(20,25,27,31)(23,28,30,32)(34,40)(36,43)(41,44)(45,47);(1,3,5,10)(2,6,9,7)(4,11,13,16)(8,14,12,15)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,35,39,38)(34,41,40,44)(36,47,43,45)(37,48,42,46);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,19)(18,22)(20,25)(21,26)(23,28)(24,29)(27,31)(30,32)(33,36,39,43)(34,42,40,37)(35,45,38,47)(41,48,44,46);(1,2)(3,7)(4,8)(5,9)(6,10)(11,15)(12,13)(14,16)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,37,39,42)(34,43,40,36)(35,46,38,48)(41,47,44,45);(1,4,5,13)(2,8,9,12)(3,11,10,16)(6,14,7,15)(17,20,21,27)(18,23,24,30)(19,25,26,31)(22,28,29,32)(33,38,39,35)(34,44,40,41)(36,47,43,45)(37,48,42,46)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=gi,d^2=g,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,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034824);