Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60753 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
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 | C22×#540 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3590 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)(7,11)(13,14,17,20,18,21,16,22)(15,19,25,26,23,27,24,28);(5,6)(7,10)(8,9)(11,12)(13,15,18,23)(14,19,21,27)(16,24,17,25)(20,26,22,28);(2,4)(6,9)(10,12)(14,20)(16,17)(19,26)(21,22)(24,25)(27,28);(5,7,8,11)(6,10,9,12)(15,23)(19,27)(24,25)(26,28);(14,21)(19,27)(20,22)(26,28)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=cf,d^b=di,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72292834, #72292835, #72292836, … |
| maximal normal subgroup of | #72292834, #72292835, #72292836, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281651);