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 | |
4030 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,11,12,8)(4,6,13,14)(5,10,15,16)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,34)(35,37)(36,42)(38,43)(39,44)(40,45)(41,48)(46,47);(1,3,7,12)(2,8,9,11)(4,10,13,16)(5,14,15,6)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,31)(29,32)(33,35)(34,41)(36,46)(37,45)(38,42)(39,47)(40,48)(43,44);(2,9)(4,13)(8,11)(10,16)(17,19,21,27)(18,22,25,23)(20,26,29,32)(24,30,28,31)(33,36,40,44)(34,42,45,39)(35,46,48,43)(37,47,41,38);(1,4)(2,6)(3,10)(5,8)(7,13)(9,14)(11,15)(12,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,30)(29,32)(33,37,40,41)(34,35,45,48)(36,38,44,47)(39,46,42,43);(1,5,7,15)(2,10,9,16)(3,6,12,14)(4,11,13,8)(17,18)(19,22)(20,28)(21,25)(23,27)(24,29)(26,31)(30,32)(33,38)(34,43)(35,42)(36,37)(39,48)(40,47)(41,44)(45,46);(1,6)(2,4)(3,5)(7,14)(8,10)(9,13)(11,16)(12,15)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,28)(27,32)(33,39,40,42)(34,44,45,36)(35,47,48,38)(37,46,41,43)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=ghi,d^2=gh,e^2=h,f^2=gh,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, 9091100);