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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2047 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)(7,10)(8,9)(11,12)(13,14,17,16,18,21,19,22)(15,23,24,28,25,27,26,20);(1,2)(3,4)(13,15)(14,20)(16,27)(17,26)(18,25)(19,24)(21,28)(22,23);(2,4)(6,9)(10,12)(13,14,19,22,18,21,17,16)(15,23,26,20,25,27,24,28);(2,4)(13,16,17,21,18,22,19,14)(15,20,24,23,25,28,26,27);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|a^2=f,c^2=fi,d^2=f,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,h^b=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6363666);