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 | |
4221 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,4,9)(5,15,16,6)(11,12,13,14)(17,18)(19,25)(20,22)(21,28)(23,27)(24,30)(26,31)(29,32)(33,34,38,41)(35,42,39,36)(37,40,46,48)(43,47,45,44);(1,3,7,4)(2,8,10,9)(5,11,16,13)(6,14,15,12)(17,19,22,27)(18,23,20,25)(21,29,30,26)(24,32,28,31)(33,35,38,39)(34,36,41,42)(37,45,46,43)(40,47,48,44);(3,4)(8,9)(11,13)(12,14)(18,20)(23,25)(24,28)(31,32)(33,34,38,41)(35,36,39,42)(37,40,46,48)(43,44,45,47);(1,4,7,3)(2,9,10,8)(5,11,16,13)(6,14,15,12)(17,20,22,18)(19,25,27,23)(21,24,30,28)(26,31,29,32)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,47)(46,48);(1,5)(2,6)(3,11)(4,13)(7,16)(8,14)(9,12)(10,15)(17,21,22,30)(18,24,20,28)(19,26,27,29)(23,31,25,32)(33,36)(34,39)(35,41)(37,44)(38,42)(40,45)(43,48)(46,47);(1,6)(2,5)(3,12)(4,14)(7,15)(8,13)(9,11)(10,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,32)(29,30)(33,37)(34,40)(35,43)(36,44)(38,46)(39,45)(41,48)(42,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=gh,d^2=hi,e^2=h,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, 8751764);