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 |
| prime radical | #43192 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4064 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)(3,11)(4,12)(5,10)(6,8)(7,13)(9,15)(14,16)(17,18,23,21)(19,26,24,20)(22,25,30,29)(27,32,31,28);(1,3,6,13)(2,7,8,11)(4,10,15,14)(5,12,16,9)(17,19,23,24)(18,20,21,26)(22,27,30,31)(25,28,29,32)(33,34,38,39)(35,41,44,40)(36,45,47,37)(42,48,46,43);(2,8)(5,16)(7,11)(9,12)(22,30)(25,29)(27,31)(28,32)(34,39)(37,45)(40,41)(43,48);(1,4)(2,9)(3,10)(5,7)(6,15)(8,12)(11,16)(13,14)(17,20,23,26)(18,24,21,19)(22,28,30,32)(25,31,29,27)(33,35,38,44)(34,40,39,41)(36,46,47,42)(37,43,45,48);(1,3)(2,7)(4,14)(5,12)(6,13)(8,11)(9,16)(10,15)(17,21,23,18)(19,20,24,26)(22,29,30,25)(27,28,31,32)(33,36)(34,37)(35,42)(38,47)(39,45)(40,48)(41,43)(44,46);(1,5,6,16)(2,10,8,14)(3,12,13,9)(4,7,15,11)(17,22,23,30)(18,25,21,29)(19,27,24,31)(20,28,26,32)(33,37)(34,36)(35,43)(38,45)(39,47)(40,46)(41,42)(44,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=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, 8300897);