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 | |
4100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18,23,24)(19,21,27,30)(20,25,29,31)(22,26,28,32)(33,34)(35,42)(36,39)(37,44)(38,41)(40,46)(43,45)(47,48);(1,2,6,7)(3,5,9,14)(4,11,13,8)(10,16,15,12)(17,19,23,27)(18,21,24,30)(20,28,29,22)(25,32,31,26)(34,36)(35,41)(40,44)(43,47);(2,7)(5,14)(8,11)(12,16)(18,24)(19,27)(22,28)(25,31)(33,35)(34,38)(36,42)(37,43)(39,41)(40,45)(44,48)(46,47);(1,3,6,9)(2,5,7,14)(4,10,13,15)(8,12,11,16)(17,20)(18,25)(19,22)(21,26)(23,29)(24,31)(27,28)(30,32)(33,36,39,34)(35,42,41,38)(37,44,46,40)(43,48,47,45);(1,4,6,13)(2,8,7,11)(3,10,9,15)(5,12,14,16)(17,21)(18,19)(20,26)(22,25)(23,30)(24,27)(28,31)(29,32)(33,37,39,46)(34,40,36,44)(35,43,41,47)(38,45,42,48);(1,5)(2,9)(3,7)(4,12)(6,14)(8,15)(10,11)(13,16)(17,22)(18,26)(19,20)(21,25)(23,28)(24,32)(27,29)(30,31)(33,38,39,42)(34,41,36,35)(37,45,46,48)(40,47,44,43)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,d^2=ghi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8135468);