Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514196 |
| 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 | #36153 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6,8,10)(7,9,11,12)(13,14)(15,18)(16,19)(17,20)(21,26)(22,28)(23,25)(24,27);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15)(14,18)(16,21)(17,22)(19,26)(20,28)(23,24)(25,27);(6,10)(7,11)(14,19)(15,21)(20,27)(22,23);(6,10)(9,12)(14,19)(15,22)(17,24)(18,25)(21,23)(26,28);(1,3,4,2)(15,21)(18,26)(22,23)(25,28)Polycyclic presentation:
2^9|a^2=gi,b^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7434871);