Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632020 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #34963 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1719 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,17)(12,20)(13,15)(14,18)(16,21)(19,22)(23,24)(25,26)(27,34)(28,32)(29,31)(30,35)(33,37)(36,39)(38,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24)(25,27,28,35)(26,30,32,34)(29,36,37,40)(31,38,33,39);(3,8)(5,7)(26,31)(29,37)(30,38)(32,33)(34,39)(36,40);(2,6)(5,7)(10,15)(11,18)(16,20)(19,24)(26,32)(27,35)(31,33)(36,40);(9,12,13,21)(10,16,15,20)(11,19,18,24)(14,22,17,23)Polycyclic presentation:
2^9|a^2=h,b^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efhi,g^a=gi,c^b=ch,d^b=dfh,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72282662, #72282830, #72283718 |
| maximal normal subgroup of | #72282662, #72282830, #72283718 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6564738);