Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667652 |
| 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 |
| prime radical | #34935 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2282 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)(13,14,18,21)(15,19,17,25)(16,23,24,27)(20,26,28,22);(2,3)(5,7)(6,9)(8,12)(10,11)(13,15,18,17)(14,19,21,25)(16,22,24,26)(20,27,28,23);(1,2,4,3)(7,12)(9,11)(13,16)(14,20)(15,22)(17,26)(18,24)(19,27)(21,28)(23,25);(6,10)(9,11)(13,15,18,17)(14,19,21,25)(16,22,24,26)(20,27,28,23);(1,3,4,2)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fh,d^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6717161);