Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438955 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2058 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)(3,4)(5,6,8,10)(7,11,12,9)(13,14,18,22)(15,24,20,16)(17,21,27,25)(19,23,26,28);(2,4)(13,15,18,20)(14,16,22,24)(17,28,27,23)(19,21,26,25);(5,7,8,12)(6,9,10,11)(13,16,26,17,18,24,19,27)(14,20,28,25,22,15,23,21);(13,17,19,16,18,27,26,24)(14,21,23,15,22,25,28,20);(14,22)(16,24)(17,27)(23,28)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gh,d^2=h,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6373220);