Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628847 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4842 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6,8,10)(7,11,12,9)(13,14)(15,18)(16,19)(17,20)(21,24)(22,26)(23,25)(27,28);(5,7,8,12)(6,9,10,11)(13,15)(14,18)(16,23)(17,22)(19,25)(20,26)(21,27)(24,28);(14,19)(16,24)(18,25)(20,21)(23,28)(26,27);(2,4)(15,22)(18,26)(23,28)(25,27);(14,20)(18,26)(19,21)(25,27)Polycyclic presentation:
2^9|a^2=f,b^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,g^a=gi,d^b=di,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72278794, #72282886, #72283217, … |
| maximal normal subgroup of | #72278794, #72282886, #72283217, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536272);