Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6668044 |
| 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 | #39340 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2117 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,10)(9,12)(15,23)(16,17)(19,26)(21,22)(24,25)(27,28);(2,3)(5,7)(6,9)(8,12)(10,11)(13,14)(15,19)(16,21)(17,22)(18,20)(23,26)(24,27)(25,28);(1,2,4,3)(7,12)(9,11)(13,15)(14,19)(16,23)(17,25)(18,24)(20,27)(21,26)(22,28);(6,10)(9,11)(14,20)(19,27)(21,22)(26,28);(1,3,4,2)(13,16,18,17)(14,21,20,22)(15,23,24,25)(19,26,27,28)Polycyclic presentation:
2^9|c^2=fh,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6717193);