Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1872 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,10)(7,11,12,9)(13,14,18,19)(15,20,24,27)(16,17,26,21)(22,23,28,25);(1,2,3,4)(13,15)(14,20)(16,25)(17,22)(18,24)(19,27)(21,28)(23,26);(5,7,8,12)(6,9,10,11)(13,16,14,21,18,26,19,17)(15,22,27,23,24,28,20,25);(13,17,19,26,18,21,14,16)(15,23,20,22,24,25,27,28);(13,14,18,19)(15,20,24,27)(16,21,26,17)(22,25,28,23)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=gh,d^2=h,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,h^b=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6373429);