Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3475 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3656 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(6,9)(10,12)(13,14)(15,19)(16,24)(17,21)(18,22)(20,23)(25,27)(26,28);(5,6)(7,10)(8,9)(11,12)(13,15,18,17)(14,19,22,21)(16,23,25,26)(20,27,28,24);(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,23)(17,26)(18,25)(19,27)(21,24)(22,28);(6,9)(10,12);(13,15,18,17)(14,21,22,19)(16,23,25,26)(20,24,28,27)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,d^b=df,g^c=gi110 conjugacy classes — view the full character table.
| maximal subgroup of | #72282528, #72282684, #72283776, … |
| maximal normal subgroup of | #72282528, #72282684, #72283776, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286495);