Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15630 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2373 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,12)(9,10)(13,14,17,16)(15,19,20,18);(5,7,9,12)(6,8,10,11)(13,15)(14,18)(16,19)(17,20)(21,22,25,24)(23,27,28,26);(6,10)(8,11)(13,15,17,20)(14,19,16,18)(21,23,25,28)(22,26,24,27);(1,2)(3,4)(5,8)(6,7)(9,11)(10,12)(13,15)(14,19)(16,18)(17,20)(21,24,25,22)(23,27,28,26);(5,7)(6,8)(9,12)(10,11)(13,16,17,14)(15,18,20,19)Polycyclic presentation:
2^9|a^2=fi,b^2=fh,c^2=h,d^2=fgh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1764204);