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 | |
3923 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)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,18,22,23)(19,21,25,29)(20,24,27,30)(26,28,31,32)(33,34,38,39)(35,36,41,44)(37,40,45,46)(42,43,47,48);(2,6)(7,9)(8,13)(14,15)(20,27)(24,30)(26,31)(28,32)(34,39)(35,41)(40,46)(42,47);(4,12)(8,13)(10,16)(14,15)(18,23)(19,25)(24,30)(26,31)(33,35)(34,36)(37,42)(38,41)(39,44)(40,43)(45,47)(46,48);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(17,19)(18,21)(20,26)(22,25)(23,29)(24,28)(27,31)(30,32)(33,36)(34,35)(37,43)(38,44)(39,41)(40,42)(45,48)(46,47);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,20,22,27)(18,24,23,30)(19,26,25,31)(21,28,29,32);(1,4,5,12)(2,8,6,13)(3,10,11,16)(7,14,9,15)(17,21)(18,19)(20,28)(22,29)(23,25)(24,26)(27,32)(30,31)(33,37,38,45)(34,40,39,46)(35,42,41,47)(36,43,44,48)Polycyclic presentation:
2^9|a^2=hi,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9388689);