Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2408 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3700 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)(6,10)(7,12)(13,14,17,18)(15,19,20,16)(21,22,24,26)(23,27,28,25);(5,6,9,10)(7,11,12,8)(13,14)(15,16)(17,18)(19,20)(21,22,24,26)(23,25,28,27);(6,10)(8,11)(14,18)(16,19)(21,22)(23,25)(24,26)(27,28);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,15)(14,19)(16,18)(17,20)(21,22)(23,25)(24,26)(27,28);(5,8,9,11)(6,7,10,12)(13,16,17,19)(14,15,18,20)(21,23)(22,25)(24,28)(26,27)Polycyclic presentation:
2^9|a^2=fi,b^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4513551);