Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437428 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2033 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,10)(7,11,12,9)(14,19)(15,17)(20,26)(21,25)(22,24)(27,28);(2,3)(13,14,18,21)(15,19,17,25)(16,20,23,27)(22,26,24,28);(1,2,4,3)(5,7,8,12)(6,9,10,11)(13,15,18,17)(14,19,21,25)(16,22,23,24)(20,26,27,28);(1,3,4,2)(13,16)(14,20)(15,22)(17,24)(18,23)(19,26)(21,27)(25,28);(16,23)(20,27)(22,24)(26,28)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=gh,d^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6373260);