Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996315 |
| 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 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2572 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)(5,6)(7,9)(8,10)(11,12)(13,14,19,23,18,21,17,24)(15,25,27,20,26,28,16,22);(5,6)(7,11)(8,10)(9,12)(13,15)(14,20)(16,19)(17,27)(18,26)(21,22)(23,25)(24,28);(2,4)(5,7)(6,9)(8,12)(10,11)(14,21)(15,26)(16,27)(23,24);(7,12)(9,11)(13,16,19,15,18,27,17,26)(14,22,23,25,21,20,24,28);(6,10)(9,11)(13,17,18,19)(14,23,21,24)(15,16,26,27)(20,28,22,25)Polycyclic presentation:
2^9|a^2=fi,d^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6903529);