Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1928 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,10)(8,9)(11,12);(2,3)(13,14,19,24)(15,21,25,28)(16,20,26,27)(17,23,18,22);(1,2,4,3)(6,9)(10,12)(13,15)(14,20)(16,17)(18,26)(19,25)(21,22)(23,28)(24,27);(1,3,4,2)(13,16,19,26)(14,21,24,28)(15,17,25,18)(20,22,27,23);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,22,24,23)(15,16,25,26)(20,21,27,28)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=f,d^2=fi,e^2=gi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6363590);