Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470092 |
| 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 | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2151 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,11)(7,12,9,10)(13,14)(15,18)(16,19)(17,20)(21,25)(22,26)(23,27)(24,28);(1,2,3,4)(5,7,8,9)(6,10,11,12)(13,15)(14,18)(16,23)(17,22)(19,27)(20,26)(21,24)(25,28);(6,10)(7,9)(11,12)(13,16,17,24)(14,19,20,28)(15,21,22,23)(18,25,26,27);(15,22)(18,26)(21,23)(25,27);(6,11)(10,12)(14,20)(18,26)(19,28)(25,27)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6384823);