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 | |
4068 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)(3,11)(4,14)(5,10)(6,7)(8,12)(9,15)(13,16)(17,18)(19,23)(20,28)(21,22)(24,30)(25,27)(26,32)(29,31)(33,34,39,42)(35,38,41,47)(36,37,40,46)(43,44,45,48);(2,7)(3,12)(9,14)(10,13)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,30)(29,31)(33,35)(34,38)(36,43)(37,44)(39,41)(40,45)(42,47)(46,48);(2,7)(5,16)(8,11)(9,14)(18,22)(23,25)(24,28)(31,32)(35,41)(38,47)(43,45)(44,48);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,19)(18,23)(20,29)(21,27)(22,25)(24,32)(26,30)(28,31)(33,36)(34,37)(35,43)(38,44)(39,40)(41,45)(42,46)(47,48);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,8,16,11)(17,20)(18,24)(19,26)(21,30)(22,28)(23,31)(25,32)(27,29)(33,37)(34,40)(35,44)(36,42)(38,45)(39,46)(41,48)(43,47);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,30)(29,32)(33,38,39,47)(34,41,42,35)(36,44,40,48)(37,45,46,43)Polycyclic presentation:
2^9|a^2=ghi,d^2=g,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8231402);