Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682460 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2349 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,10)(9,12)(13,14)(15,20)(16,26)(17,22)(18,24)(19,23)(21,28)(25,27);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,22)(17,26)(18,25)(19,21)(23,28)(24,27);(2,4)(13,16,18,28,19,27,17,20)(14,21,22,26,23,15,24,25);(6,10)(9,11)(15,21)(16,27)(20,28)(25,26);(2,4)(7,12)(9,11)(13,17,19,18)(14,22,23,24)(15,25,21,26)(16,20,27,28)Polycyclic presentation:
2^9|c^2=g,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6589666);