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 | #57387 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3952 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,4,9)(5,15,16,6)(11,12,13,14)(17,18,23,20)(19,26,24,21)(22,28,30,25)(27,29,31,32)(33,34)(35,44)(36,37)(38,46)(39,40)(41,45)(42,48)(43,47);(1,3,7,4)(2,8,10,9)(5,11,16,13)(6,14,15,12)(19,24)(21,26)(22,30)(25,28)(34,40)(35,45)(37,42)(43,46);(3,4)(8,9)(11,13)(12,14)(17,19,23,24)(18,21,20,26)(22,27,30,31)(25,29,28,32)(33,35)(34,41)(36,46)(37,47)(38,42)(39,45)(40,44)(43,48);(1,4,7,3)(2,9,10,8)(5,11,16,13)(6,14,15,12)(17,20,23,18)(19,21,24,26)(22,28,30,25)(27,29,31,32)(33,36)(34,42)(35,46)(37,40)(38,41)(39,48)(43,45)(44,47);(1,5)(2,6)(3,11)(4,13)(7,16)(8,14)(9,12)(10,15)(17,21)(18,24)(19,20)(22,29)(23,26)(25,31)(27,28)(30,32)(33,37,39,42)(34,36,40,48)(35,47,45,38)(41,46,44,43);(1,6)(2,5)(3,12)(4,14)(7,15)(8,13)(9,11)(10,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,38,39,47)(34,43,40,46)(35,42,45,37)(36,41,48,44)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=gh,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8751050);