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 | |
4534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,17)(12,20)(13,18)(14,15)(16,21)(19,23)(22,24)(25,26,30,33)(27,28,31,35)(29,32,38,40)(34,36,37,39);(2,3)(6,7)(10,15)(12,21)(13,17)(19,22)(25,27)(26,28)(29,37)(30,31)(32,39)(33,35)(34,38)(36,40);(1,3,5,2)(4,7,8,6)(10,15)(11,18)(16,20)(19,22)(27,31)(28,35)(34,37)(36,39);(1,4,5,8)(2,6,3,7)(9,11)(10,13)(12,22)(14,18)(15,17)(16,24)(19,21)(20,23)(25,28,30,35)(26,31,33,27)(29,36,38,39)(32,37,40,34);(9,12)(10,16)(11,19)(13,23)(14,21)(15,20)(17,24)(18,22)(25,26)(27,28)(29,32)(30,33)(31,35)(34,36)(37,39)(38,40);(9,13)(10,11)(12,23)(14,17)(15,18)(16,19)(20,22)(21,24)(25,29)(26,32)(27,34)(28,36)(30,38)(31,37)(33,40)(35,39)Polycyclic presentation:
2^9|a^2=i,c^2=gh,d^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8746577);