Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514172 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
2665 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)(11,21)(12,25)(13,20)(14,19)(15,18)(16,24)(17,27)(22,38)(23,33)(26,37)(28,36)(29,32)(30,35)(31,39)(34,40);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,26)(13,23)(14,24)(15,29)(17,34)(18,33)(19,21)(20,32)(22,27)(25,31)(28,39)(30,40)(35,38)(36,37);(9,12,14,28)(10,17,19,35)(11,22,24,40)(13,27,15,30)(16,31,21,37)(18,25,20,36)(23,39,29,26)(32,34,33,38);(10,18)(13,15)(16,32)(17,25)(19,20)(21,33)(23,29)(26,39)(27,30)(31,34)(35,36)(37,38);(2,6)(5,7)(9,13,14,15)(10,18,19,20)(11,23,24,29)(12,27,28,30)(16,33,21,32)(17,25,35,36)(22,39,40,26)(31,38,37,34)Polycyclic presentation:
2^9|b^2=fi,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,g^c=gi,h^c=hi,e^d=ei,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421557);