Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437753 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#2619 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,9)(7,11,12,10)(13,14,19,22)(15,17,21,18)(16,20,24,27)(23,25,28,26);(1,2)(3,4)(6,9)(10,11)(13,15,18,14,19,21,17,22)(16,23,26,20,24,28,25,27);(2,4)(5,7,8,12)(6,10,9,11)(13,16,19,24)(14,20,22,27)(15,23,21,28)(17,25,18,26);(2,4)(16,24)(20,27)(23,28)(25,26);(13,17,19,18)(14,21,22,15)(16,25,24,26)(20,28,27,23)Polycyclic presentation:
2^9|a^2=gi,b^2=fg,c^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6354837);