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 | |
4334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,8)(3,6,11,9)(4,12,14,7)(10,16,13,15)(17,18)(19,24)(20,22)(21,28)(23,27)(25,29)(26,30)(31,32)(33,34,40,38)(35,37,41,36)(39,46,47,42)(43,44,48,45);(1,2,5,8)(3,6,11,9)(4,7,14,12)(10,15,13,16)(17,19)(18,23)(20,24)(21,26)(22,27)(25,30)(28,31)(29,32)(33,35)(34,36)(37,38)(39,43)(40,41)(42,44)(45,46)(47,48);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,14)(13,16)(17,18,22,20)(19,24,27,23)(21,25,29,28)(26,31,32,30)(33,36,40,37)(34,41,38,35)(39,44,47,45)(42,48,46,43);(1,2)(3,9)(4,7)(5,8)(6,11)(10,16)(12,14)(13,15)(17,20,22,18)(19,24,27,23)(21,28,29,25)(26,31,32,30)(33,37,40,36)(34,41,38,35)(39,45,47,44)(42,48,46,43);(1,3,5,11)(2,6,8,9)(4,13,14,10)(7,16,12,15)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32)(33,38,40,34)(35,37,41,36)(39,42,47,46)(43,44,48,45);(1,4)(2,7)(3,10)(5,14)(6,15)(8,12)(9,16)(11,13)(17,21,22,29)(18,25,20,28)(19,26,27,32)(23,30,24,31)(33,39,40,47)(34,42,38,46)(35,43,41,48)(36,44,37,45)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=g,d^2=g,e^2=gi,f^2=g,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, 10411797);