Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1418353 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2450 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)(6,7)(8,12)(14,17)(15,19)(21,22,24,26)(23,27,28,25);(5,6,9,7)(8,11,12,10)(23,28)(25,27);(5,6,9,7)(8,10,12,11)(13,14)(15,18)(16,17)(19,20)(21,23)(22,25)(24,28)(26,27);(1,2)(3,4)(5,7,9,6)(8,10,12,11)(13,15,16,19)(14,18,17,20);(5,8,9,12)(6,10,7,11)(13,14)(15,18)(16,17)(19,20)(21,22)(23,25)(24,26)(27,28)Polycyclic presentation:
2^9|a^2=gi,b^2=f,c^2=f,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1337377);