Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6629009 |
| 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×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2416 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)(14,18)(15,22)(16,19)(20,27)(21,25)(23,26);(6,9)(10,12)(13,14,19,25,17,21,16,18)(15,20,26,28,23,27,22,24);(5,7,8,11)(6,10,9,12)(13,15,16,22,17,23,19,26)(14,20,18,24,21,27,25,28);(2,4)(14,21)(18,25)(20,27)(24,28);(14,21)(15,23)(18,25)(22,26)Polycyclic presentation:
2^9|b^2=gi,c^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,d^b=di,e^b=ei,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72282833, #72283339, #72283728, … |
| maximal normal subgroup of | #72282833, #72283339, #72283728, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536693);