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 | |
4095 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,6)(3,9,4,8)(5,10,13,14)(11,16,12,15)(17,18,21,25)(19,24,27,22)(20,28,29,23)(26,30,32,31)(33,34,39,44)(35,40,46,37)(36,42,48,41)(38,47,45,43);(1,3,7,4)(2,8,6,9)(5,12,13,11)(10,16,14,15)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32)(33,35,39,46)(34,40,44,37)(36,45,48,38)(41,47,42,43);(17,20,21,29)(18,23,25,28)(19,26,27,32)(22,30,24,31)(33,36,39,48)(34,41,44,42)(35,38,46,45)(37,47,40,43);(1,4,7,3)(2,9,6,8)(5,12,13,11)(10,16,14,15)(17,20)(18,23)(19,26)(21,29)(22,30)(24,31)(25,28)(27,32)(33,36)(34,42)(35,45)(37,47)(38,46)(39,48)(40,43)(41,44);(1,5,7,13)(2,10,6,14)(3,11,4,12)(8,15,9,16)(17,19)(18,24)(20,26)(21,27)(22,25)(23,31)(28,30)(29,32)(33,37)(34,35)(36,47)(38,41)(39,40)(42,45)(43,48)(44,46);(1,6,7,2)(3,9,4,8)(5,10,13,14)(11,15,12,16)(17,18)(19,22)(20,23)(21,25)(24,27)(26,30)(28,29)(31,32)(33,38,39,45)(34,43,44,47)(35,36,46,48)(37,42,40,41)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=h,d^2=ghi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9770604);