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 | |
4056 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,4)(3,8,6,10)(5,9,14,12)(11,15,13,16)(17,18)(19,23)(20,28)(21,22)(24,30)(25,27)(26,32)(29,31)(33,34)(35,39)(36,43)(37,42)(38,48)(40,45)(41,47)(44,46);(1,3,7,6)(2,8,4,10)(5,11,14,13)(9,15,12,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,30)(29,31)(33,35)(34,41)(36,46)(37,38)(39,45)(40,47)(42,44)(43,48);(3,6)(8,10)(11,13)(15,16)(18,22)(23,25)(24,28)(31,32)(33,36,40,37)(34,42,45,43)(35,46,47,38)(39,48,41,44);(1,4,7,2)(3,10,6,8)(5,9,14,12)(11,15,13,16)(17,19)(18,23)(20,29)(21,27)(22,25)(24,32)(26,30)(28,31)(33,37,40,36)(34,43,45,42)(35,46,47,38)(39,48,41,44);(1,5,7,14)(2,9,4,12)(3,11,6,13)(8,15,10,16)(17,20)(18,24)(19,26)(21,30)(22,28)(23,31)(25,32)(27,29)(33,38,40,46)(34,44,45,48)(35,37,47,36)(39,43,41,42);(1,6,7,3)(2,10,4,8)(5,13,14,11)(9,16,12,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,30)(29,32)(33,39,40,41)(34,35,45,47)(36,44,37,48)(38,43,46,42)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=g,d^2=hi,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8253208);