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 | |
4011 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,3)(7,9)(8,10)(14,15)(17,18,21,22)(19,23,27,25)(20,28,29,24)(26,31,32,30);(1,2,6,3)(4,7,11,9)(5,8,13,10)(12,14,16,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,34,38,39)(35,37,41,46)(36,40,43,47)(42,45,44,48);(4,11)(5,13)(7,9)(8,10)(18,22)(23,25)(24,28)(30,31)(34,39)(36,43)(37,46)(42,44);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,35,38,41)(34,37,39,46)(36,44,43,42)(40,48,47,45);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31)(33,36)(34,40)(35,42)(37,45)(38,43)(39,47)(41,44)(46,48);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)(33,37)(34,41)(35,39)(36,45)(38,46)(40,44)(42,47)(43,48)Polycyclic presentation:
2^9|a^2=h,b^2=gi,d^2=gi,e^2=g,f^2=gh,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, 9416016);