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 | |
4133 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,12)(3,13,16,8)(4,6,11,9)(5,10,14,15)(33,34)(35,42)(36,44)(37,41)(38,39)(40,46)(43,48)(45,47);(1,3,7,16)(2,8,12,13)(4,5,11,14)(6,15,9,10)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,31)(30,32)(33,35)(34,37)(36,45)(38,42)(39,41)(40,47)(43,46)(44,48);(4,11)(5,14)(6,9)(10,15)(18,22)(19,26)(20,29)(31,32)(34,38)(36,46)(37,42)(43,45);(1,4,7,11)(2,9,12,6)(3,14,16,5)(8,15,13,10)(17,19)(18,23)(20,30)(21,26)(22,25)(24,32)(27,29)(28,31)(33,36)(34,40)(35,43)(37,48)(38,44)(39,46)(41,45)(42,47);(1,5,7,14)(2,10,12,15)(3,4,16,11)(6,8,9,13)(17,20)(18,24)(19,27)(21,29)(22,28)(23,31)(25,32)(26,30)(33,37,39,42)(34,41,38,35)(36,47,46,48)(40,43,44,45);(1,6,7,9)(2,11,12,4)(3,15,16,10)(5,8,14,13)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,29)(30,32)(33,38,39,34)(35,42,41,37)(36,44,46,40)(43,47,45,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,d^2=gh,e^2=g,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, 8965721);