Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3335 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,8)(13,14,18,19)(15,20,23,27)(16,17,22,21)(24,26,28,25);(1,2)(3,4)(9,10,11,12)(14,19)(15,23)(16,22)(25,26);(1,2,3,4)(9,10)(11,12)(13,15)(14,20)(16,24)(17,26)(18,23)(19,27)(21,25)(22,28);(5,6,7,8)(13,16,18,22)(14,21,19,17)(15,24,23,28)(20,25,27,26);(1,2)(3,4)(13,17,18,21)(14,22,19,16)(15,25,23,26)(20,24,27,28)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=g,d^2=i,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72280828, #72280957, #72281051, … |
| maximal normal subgroup of | #72280828, #72280957, #72281051, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2213657);