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 | |
4114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,8)(4,10)(5,11)(6,14)(7,9)(12,16)(13,15)(18,23)(19,25)(24,30)(27,31);(1,3)(2,8)(4,7)(5,12)(6,13)(9,10)(11,16)(14,15)(17,18)(19,26)(20,25)(21,24)(22,23)(27,32)(28,31)(29,30)(33,34)(35,41)(36,44)(37,38)(39,42)(40,45)(43,47)(46,48);(2,9)(3,4)(5,15)(6,16)(19,25)(20,26)(21,29)(24,30)(34,38)(35,42)(36,45)(47,48);(1,4,7,3)(2,10,9,8)(5,12,15,14)(6,11,16,13)(17,19,22,25)(18,20,23,26)(21,27,29,31)(24,28,30,32)(33,35)(34,39)(36,46)(37,42)(38,41)(40,48)(43,45)(44,47);(1,5,7,15)(2,11,9,13)(3,12,4,14)(6,8,16,10)(17,20)(18,19)(21,28)(22,26)(23,25)(24,27)(29,32)(30,31)(33,36)(34,40)(35,43)(37,45)(38,44)(39,47)(41,48)(42,46);(1,6)(2,12)(3,13)(4,11)(5,8)(7,16)(9,14)(10,15)(17,21,22,29)(18,24,23,30)(19,27,25,31)(20,28,26,32)(33,34)(35,39)(36,40)(37,38)(41,42)(43,47)(44,45)(46,48)Polycyclic presentation:
2^9|d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8966288);