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 | |
4028 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,8,11,15)(4,9,14,13)(5,12,16,10)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,34)(35,44)(36,40)(37,41)(38,42)(39,43)(45,46)(47,48);(2,7)(4,14)(8,15)(10,12)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,31)(29,32)(33,35,39,42)(34,38,43,44)(36,45,41,48)(37,46,40,47);(2,7)(3,11)(9,13)(10,12)(17,19,21,27)(18,22,25,23)(20,26,29,32)(24,30,28,31)(33,36,39,41)(34,40,43,37)(35,45,42,48)(38,47,44,46);(1,3,6,11)(2,8,7,15)(4,10,14,12)(5,13,16,9)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,30)(29,32)(33,36)(34,40)(35,45)(37,43)(38,47)(39,41)(42,48)(44,46);(1,4)(2,9)(3,12)(5,15)(6,14)(7,13)(8,16)(10,11)(17,18)(19,22)(20,28)(21,25)(23,27)(24,29)(26,31)(30,32)(33,37,39,40)(34,41,43,36)(35,46,42,47)(38,48,44,45);(1,5,6,16)(2,10,7,12)(3,13,11,9)(4,8,14,15)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,28)(27,32)(33,38)(34,42)(35,43)(36,47)(37,45)(39,44)(40,48)(41,46)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=ghi,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9091082);