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 | |
4113 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,8)(3,5,4,11)(6,9,15,16)(10,13,12,14)(17,18,23,25)(19,27,21,24)(20,26,30,31)(22,28,32,29)(33,34,39,43)(35,40,38,42)(36,41,47,37)(44,46,45,48);(1,3)(2,5)(4,7)(6,10)(8,11)(9,13)(12,15)(14,16)(17,19,23,21)(18,24,25,27)(20,29,30,28)(22,26,32,31)(33,35,39,38)(34,40,43,42)(36,45,47,44)(37,46,41,48);(2,8)(3,4)(9,16)(10,12)(18,25)(20,30)(24,27)(28,29)(33,36,39,47)(34,41,43,37)(35,44,38,45)(40,46,42,48);(1,4,7,3)(2,5,8,11)(6,10,15,12)(9,14,16,13)(17,20)(18,26)(19,28)(21,29)(22,24)(23,30)(25,31)(27,32)(33,36)(34,37)(35,45)(38,44)(39,47)(40,46)(41,43)(42,48);(1,5)(2,3)(4,8)(6,14)(7,11)(9,12)(10,16)(13,15)(17,21,23,19)(18,27,25,24)(20,29,30,28)(22,26,32,31)(33,37)(34,36)(35,46)(38,48)(39,41)(40,45)(42,44)(43,47);(1,6,7,15)(2,9,8,16)(3,10,4,12)(5,13,11,14)(17,22,23,32)(18,28,25,29)(19,26,21,31)(20,27,30,24)(33,38,39,35)(34,42,43,40)(36,45,47,44)(37,46,41,48)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=ghi,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8657228);