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 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1932 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)(5,6,8,10)(7,11,12,9)(14,20)(21,22)(23,24)(25,27);(2,3)(13,14,18,20)(15,25,16,27)(17,24,19,23)(21,28,22,26);(1,2,4,3)(5,7,8,12)(6,9,10,11)(13,15,18,16)(14,21,20,22)(17,26,19,28)(23,25,24,27);(1,3,4,2)(13,16,18,15)(14,22,20,21)(17,28,19,26)(23,27,24,25);(13,17,18,19)(14,23,20,24)(15,26,16,28)(21,25,22,27)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=fg,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,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, 6353292);