Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438967 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2026 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,20)(17,21)(19,24)(22,25)(23,27)(26,28);(1,2,3,4)(15,22)(16,23)(18,25)(19,26)(20,27)(24,28);(6,9)(10,12)(13,15,16,22,17,24,23,28)(14,18,20,25,21,19,27,26);(13,15,23,28,17,24,16,22)(14,19,27,25,21,18,20,26);(5,7,8,11)(6,10,9,12)(15,24)(18,19)(22,28)(25,26)Polycyclic presentation:
2^9|b^2=f,c^2=h,d^2=hi,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72283139, #72283890, #72284894, … |
| maximal normal subgroup of | #72283139, #72283890, #72284894, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374901);