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 | |
4488 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(4,8)(6,7)(9,10)(11,17)(12,18)(13,16)(14,15)(19,23)(20,24)(21,22)(25,26)(27,32)(28,37)(29,39)(30,31)(33,38)(34,35)(36,40);(2,3)(6,7)(10,15)(11,18)(16,22)(19,24)(26,31)(32,40)(33,37)(34,39);(1,2,5,3)(4,6,8,7)(9,11)(10,12)(13,19)(14,18)(15,17)(16,20)(21,24)(22,23)(25,27,30,36)(26,32,31,40)(28,35,38,29)(33,39,37,34);(9,11)(10,12)(13,19)(14,18)(15,17)(16,20)(21,24)(22,23)(25,28)(26,33)(27,35)(29,36)(30,38)(31,37)(32,39)(34,40);(1,3,5,2)(4,7,8,6)(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,23)(22,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(1,4)(2,6)(3,7)(5,8)(9,13,14,21)(10,16,15,22)(11,19,18,24)(12,20,17,23)(25,29,30,35)(26,34,31,39)(27,28,36,38)(32,33,40,37)Polycyclic presentation:
2^9|c^2=gi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8017189);