Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432838 |
| 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 | |
3064 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(7,11)(10,12)(13,14)(15,24)(16,25)(17,23)(18,22)(19,21)(20,27)(26,28);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,25)(18,26)(19,20)(21,28)(22,24)(23,27);(6,9)(10,12)(13,16)(14,20)(15,21)(17,24)(18,27)(19,28)(22,26)(23,25);(5,7,8,11)(6,10,9,12)(14,21)(15,20)(22,23)(25,26);(13,17,19,18)(14,22,21,23)(15,25,20,26)(16,24,28,27)Polycyclic presentation:
2^9|a^2=fg,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=di,f^b=fi,g^b=gi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342640);