Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #156684 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
3020 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,7,8)(9,10,11,12)(13,14)(15,16)(17,18)(19,20);(2,4)(14,18)(16,19)(21,22,24,26)(23,27,28,25);(9,10)(11,12)(13,15)(14,19)(16,18)(17,20)(21,23,24,28)(22,25,26,27);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,16,17,19)(14,15,18,20)(21,22)(23,25)(24,26)(27,28)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72267675, #72267684, #72267685, … |
| maximal normal subgroup of | #72267675, #72267684, #72267685, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 64331);