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 | |
4136 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)(3,4)(5,8)(6,9)(7,10)(11,12)(13,15)(14,16)(17,18,22,25)(19,26,20,23)(21,29,30,24)(27,31,28,32)(33,34,38,43)(35,42,45,39)(36,46,47,40)(37,41,44,48);(1,3,7,5)(2,8,10,4)(6,11,14,13)(9,15,16,12)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,31)(29,30)(33,35)(34,39)(36,44)(37,47)(38,45)(40,48)(41,46)(42,43);(17,19,22,20)(18,23,25,26)(21,27,30,28)(24,31,29,32)(33,36,38,47)(34,40,43,46)(35,44,45,37)(39,48,42,41);(1,4,7,8)(2,3,10,5)(6,12,14,15)(9,11,16,13)(17,18)(19,23)(20,26)(21,24)(22,25)(27,31)(28,32)(29,30)(33,37)(34,41)(35,36)(38,44)(39,40)(42,46)(43,48)(45,47);(1,5,7,3)(2,8,10,4)(6,11,14,13)(9,12,16,15)(17,20,22,19)(18,23,25,26)(21,28,30,27)(24,31,29,32)(33,35)(34,42)(36,44)(37,47)(38,45)(39,43)(40,41)(46,48);(1,6,7,14)(2,9,10,16)(3,11,5,13)(4,12,8,15)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,34)(35,39)(36,40)(37,41)(38,43)(42,45)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=h,d^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10411690);