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 | |
4167 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,9,10,7)(5,12,13,8)(11,14,16,15)(18,19)(23,25)(24,26)(30,31)(33,34,38,40)(35,39,44,43)(36,41,45,46)(37,42,47,48);(2,4)(7,9)(8,12)(14,15)(17,18,22,19)(20,23,27,25)(21,24,29,26)(28,30,32,31)(33,35)(34,39)(36,42)(37,46)(38,44)(40,43)(41,47)(45,48);(2,4)(3,10)(5,13)(14,15)(20,27)(21,29)(23,25)(24,26)(34,40)(35,44)(36,45)(37,47);(1,3,6,10)(2,7,4,9)(5,11,13,16)(8,14,12,15)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,30,32,31)(33,36,38,45)(34,41,40,46)(35,42,44,48)(37,39,47,43);(1,4,6,2)(3,7,10,9)(5,12,13,8)(11,14,16,15)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,37)(34,42)(35,41)(36,43)(38,47)(39,45)(40,48)(44,46);(1,5,6,13)(2,8,4,12)(3,11,10,16)(7,14,9,15)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(33,35)(34,43)(36,42)(37,41)(38,44)(39,40)(45,48)(46,47)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,d^2=gi,e^2=g,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^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9414096);