Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #845297 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,10,11)(7,9,8,12)(13,14,17,15)(16,19,20,18)(22,26)(24,28);(6,11)(7,8)(14,15)(18,19)(21,22,25,26)(23,24,27,28);(5,7,10,8)(6,12,11,9)(13,15,17,14)(16,19,20,18)(21,23,25,27)(22,24,26,28);(1,2)(3,4)(5,8,10,7)(6,9,11,12);(5,9)(6,7)(8,11)(10,12)(13,16)(14,18)(15,19)(17,20)(21,24)(22,27)(23,26)(25,28)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=f,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg92 conjugacy classes — view the full character table.
| maximal subgroup of | #72327933, #72327939, #72327997, … |
| maximal normal subgroup of | #72327933, #72327939, #72327997, … |
| maximal quotient of | #72329952, #72330126, #72330433, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 753664);