Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393944 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,3,4)(5,6)(7,8)(9,10,13,15)(11,16,12,14)(18,22)(19,25)(24,29)(27,30);(9,11,13,12)(10,14,15,16)(17,18,20,22)(19,26,25,32)(21,29,28,24)(23,30,31,27);(2,4)(17,19,20,25)(18,23,22,31)(21,30,28,27)(24,32,29,26);(6,8)(9,12,13,11)(10,14,15,16)(19,25)(23,31)(26,32)(27,30)Polycyclic presentation:
2^9|a^2=ef,b^2=e,c^2=i,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,d^c=di,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72264138, #72264172, #72264335, … |
| maximal normal subgroup of | #72264138, #72264172, #72264335, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303608);