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 | |
4123 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,9,10)(4,12,13,8)(11,16,15,14)(17,18)(19,27)(20,25)(21,26)(22,23)(24,28)(29,32)(30,31)(33,34,38,40)(35,39,36,41)(37,42,45,47)(43,46,44,48);(2,6)(7,10)(8,12)(14,16)(18,23)(19,28)(20,26)(30,32)(33,35)(34,39)(36,38)(37,43)(40,41)(42,46)(44,45)(47,48);(2,6)(3,9)(4,13)(14,16)(19,28)(20,26)(21,25)(24,27)(34,40)(35,36)(37,45)(46,48);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,19,22,28)(18,24,23,27)(20,31,26,29)(21,30,25,32)(33,36,38,35)(34,41,40,39)(37,43,45,44)(42,46,47,48);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,20,22,26)(18,25,23,21)(19,29,28,31)(24,30,27,32)(33,37,38,45)(34,42,40,47)(35,43,36,44)(39,46,41,48);(1,4,5,13)(2,8,6,12)(3,11,9,15)(7,14,10,16)(17,21,22,25)(18,26,23,20)(19,30,28,32)(24,29,27,31)(33,34)(35,39)(36,41)(37,42)(38,40)(43,46)(44,48)(45,47)Polycyclic presentation:
2^9|a^2=h,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8966356);