Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432861 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2934 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)(7,11)(10,12)(14,15)(16,24)(17,18)(20,27)(21,22)(23,28);(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,19,22)(15,17,21,18)(16,20,24,28)(23,25,27,26);(6,9)(10,12)(13,15,19,21)(14,18,22,17)(16,23,24,27)(20,26,28,25);(5,7,8,11)(6,10,9,12)(13,16,19,24)(14,20,22,28)(15,23,21,27)(17,25,18,26);(16,24)(20,28)(23,27)(25,26)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=i,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342653);