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 | |
4305 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,11,9)(4,12,13,8)(10,15,16,14)(17,18,22,24)(19,20,27,23)(21,29,31,25)(26,32,30,28)(33,34,39,37)(35,40,41,36)(38,45,46,42)(43,44,48,47);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19,22,27)(18,23,24,20)(21,26,31,30)(25,32,29,28)(33,35)(34,40)(36,37)(38,43)(39,41)(42,47)(44,45)(46,48);(2,6)(7,9)(8,12)(14,15)(18,24)(20,23)(25,29)(28,32)(33,36,39,40)(34,41,37,35)(38,44,46,47)(42,48,45,43);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20,22,23)(18,19,24,27)(21,28,31,32)(25,26,29,30)(33,36)(34,35)(37,41)(38,44)(39,40)(42,43)(45,48)(46,47);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,15)(13,16)(17,19)(18,20)(21,30)(22,27)(23,24)(25,32)(26,31)(28,29)(33,37,39,34)(35,40,41,36)(38,42,46,45)(43,44,48,47);(1,4,5,13)(2,8,6,12)(3,10,11,16)(7,14,9,15)(17,21,22,31)(18,25,24,29)(19,26,27,30)(20,28,23,32)(33,38)(34,42)(35,43)(36,44)(37,45)(39,46)(40,47)(41,48)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=gh,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154542);