Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500908 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2642 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,18,15,17,32,35)(11,13,21,28,20,30,22,29)(12,23,26,34,27,36,31,16)(19,37,39,40,38,33,24,25);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,24)(14,21)(15,20)(16,33)(17,30)(18,28)(19,26)(22,32)(23,25)(27,39)(29,35)(31,38)(34,37)(36,40);(9,12,15,27)(10,16,17,34)(11,19,20,38)(13,25,30,40)(14,31,32,26)(18,36,35,23)(21,24,22,39)(28,33,29,37);(2,6)(3,8)(9,13)(10,11)(12,25)(14,29)(15,30)(16,19)(17,20)(18,22)(21,35)(23,24)(26,33)(27,40)(28,32)(31,37)(34,38)(36,39);(10,17)(11,20)(16,34)(18,35)(19,38)(21,22)(23,36)(24,39)Polycyclic presentation:
2^9|a^2=g,c^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,h^b=hi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410458);