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 | |
4293 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)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,30)(31,32)(33,34,39,41)(35,38,36,40)(37,42,46,48)(43,45,44,47);(3,9)(7,13)(10,15)(14,16)(18,22)(20,29)(23,25)(27,32)(33,35,39,36)(34,40,41,38)(37,44,46,43)(42,45,48,47);(2,6)(7,13)(8,11)(14,16)(18,22)(19,26)(24,28)(27,32)(34,41)(35,36)(42,48)(43,44);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19,21,26)(18,23,22,25)(20,27,29,32)(24,30,28,31)(33,36,39,35)(34,38,41,40)(37,44,46,43)(42,45,48,47);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32)(33,37)(34,42)(35,43)(36,44)(38,45)(39,46)(40,47)(41,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,38)(34,35)(36,41)(37,45)(39,40)(42,43)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9150794);