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 | |
4189 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,4,10,8)(5,13,14,9)(11,15,16,12)(20,27)(24,29)(25,31)(30,32)(33,34,38,40)(35,39,42,36)(37,45,46,41)(43,44,48,47);(2,7)(3,10)(9,13)(11,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)(33,35)(34,39)(36,40)(37,43)(38,42)(41,47)(44,45)(46,48);(2,7)(4,8)(9,13)(12,15)(18,22)(23,28)(24,29)(30,32)(34,40)(35,42)(41,45)(43,48);(1,3)(2,8)(4,7)(5,11)(6,10)(9,15)(12,13)(14,16)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,36)(34,35)(37,44)(38,39)(40,42)(41,43)(45,48)(46,47);(1,4,6,8)(2,3,7,10)(5,12,14,15)(9,11,13,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,47)(46,48);(1,5,6,14)(2,9,7,13)(3,11,10,16)(4,12,8,15)(17,20)(18,24)(19,25)(21,27)(22,29)(23,30)(26,31)(28,32)(33,37,38,46)(34,41,40,45)(35,43,42,48)(36,44,39,47)Polycyclic presentation:
2^9|a^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, 8025800);