Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437753 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3455 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3088 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)(7,10,11,12)(13,14,19,22)(15,17,21,18)(16,24,28,20)(23,26,25,27);(1,2)(3,4)(13,15,18,14,19,21,17,22)(16,25,26,24,28,23,27,20);(2,4)(6,9)(10,12)(13,16)(14,20)(15,23)(17,27)(18,26)(19,28)(21,25)(22,24);(2,4)(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,21,22,15)(16,26,28,27)(20,23,24,25);(13,17,19,18)(14,21,22,15)(16,27,28,26)(20,25,24,23)Polycyclic presentation:
2^9|a^2=g,b^2=f,d^2=g,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, 6357259);