Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438967 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2470 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)(7,11)(8,9)(10,12)(13,14,16,21,17,20,24,18)(15,19,22,26,23,27,28,25);(2,4)(6,9)(10,11)(14,18)(15,22)(16,24)(20,21)(23,28)(25,26);(5,7,8,12)(6,10,9,11)(13,15,16,22,17,23,24,28)(14,19,21,26,20,27,18,25);(13,15,16,22,17,23,24,28)(14,19,21,26,20,27,18,25);(14,20)(15,23)(18,21)(22,28)Polycyclic presentation:
2^9|a^2=fg,c^2=gh,d^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72316273 |
| maximal normal subgroup of | #72316273 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6373404);