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 | |
4133 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)(3,5,9,15)(4,8,13,16)(10,14,11,12)(17,18)(19,28)(20,30)(21,27)(22,24)(23,25)(26,31)(29,32)(33,34)(35,44)(36,41)(37,47)(38,42)(39,43)(40,46)(45,48);(1,3)(2,5)(4,11)(6,9)(7,15)(8,12)(10,13)(14,16)(17,19,23,22)(18,24,25,28)(20,29,31,27)(21,26,32,30)(33,35)(34,40)(36,38)(37,45)(39,46)(41,48)(42,47)(43,44);(2,7)(3,9)(8,16)(10,11)(18,25)(20,31)(24,28)(27,29)(33,36,39,37)(34,41,43,47)(35,38,46,45)(40,48,44,42);(1,3)(2,5)(4,11)(6,9)(7,15)(8,12)(10,13)(14,16)(17,20)(18,26)(19,29)(21,28)(22,27)(23,31)(24,32)(25,30)(33,37,39,36)(34,41,43,47)(35,45,46,38)(40,48,44,42);(1,4,6,13)(2,8,7,16)(3,10,9,11)(5,14,15,12)(17,21,23,32)(18,27,25,29)(19,30,22,26)(20,24,31,28)(33,34)(35,40)(36,41)(37,47)(38,48)(39,43)(42,45)(44,46);(1,5)(2,3)(4,12)(6,15)(7,9)(8,11)(10,16)(13,14)(17,22,23,19)(18,28,25,24)(20,27,31,29)(21,30,32,26)(33,38,39,45)(34,42,43,48)(35,36,46,37)(40,47,44,41)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=ghi,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10049367);