Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2299 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)(5,6)(7,10)(8,9)(11,12)(13,14)(15,21)(16,20)(17,22)(18,23)(19,24)(25,27)(26,28);(2,4)(13,15,18,16)(14,20,23,21)(17,19,26,25)(22,27,28,24);(6,9)(10,12)(13,16,19,26,18,15,25,17)(14,21,24,22,23,20,27,28);(13,17,19,16,18,26,25,15)(14,22,24,20,23,28,27,21);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=h,e^2=gi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72283931 |
| maximal normal subgroup of | #72283931 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374463);