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 | #37148 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3953 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,9,11,7)(5,12,14,8)(10,15,13,16)(17,18,22,20)(19,23,25,27)(21,24,30,28)(26,31,29,32)(33,34,40,43)(35,38,42,48)(36,45,47,41)(37,39,46,44);(1,3,6,11)(2,7,4,9)(5,13,14,10)(8,16,12,15)(17,19)(18,23)(20,27)(21,29)(22,25)(24,32)(26,30)(28,31)(33,35,40,42)(34,38,43,48)(36,44,47,39)(37,41,46,45);(2,4)(7,9)(8,12)(15,16)(19,25)(23,27)(26,29)(31,32)(33,36,40,47)(34,41,43,45)(35,44,42,39)(37,38,46,48);(1,3)(2,7)(4,9)(5,10)(6,11)(8,15)(12,16)(13,14)(17,20,22,18)(19,23,25,27)(21,28,30,24)(26,31,29,32)(33,37)(34,39)(35,41)(36,38)(40,46)(42,45)(43,44)(47,48);(1,4,6,2)(3,9,11,7)(5,12,14,8)(10,16,13,15)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,32)(29,30)(33,38)(34,42)(35,43)(36,46)(37,47)(39,41)(40,48)(44,45);(1,5,6,14)(2,8,4,12)(3,10,11,13)(7,15,9,16)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,29)(27,32)(33,39)(34,37)(35,36)(38,41)(40,44)(42,47)(43,46)(45,48)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=gh,d^2=i,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10259717);