Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996671 |
| 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 | #24439 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1914 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14,18,27,17,21,16,20)(15,22,23,19,26,24,25,28);(5,6)(7,11)(8,10)(9,12)(13,15,17,26)(14,19,21,28)(16,23,18,25)(20,24,27,22);(2,4)(5,7,8,12)(6,9,10,11)(13,16,17,18)(14,20,21,27)(15,23,26,25)(19,24,28,22);(7,12)(9,11)(13,14,16,20,17,21,18,27)(15,24,25,19,26,22,23,28);(6,10)(9,11)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=hi,d^2=f,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6972167);