Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437418 |
| 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×#3494 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3336 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)(7,10)(8,11)(9,12)(14,19)(15,23)(17,24)(20,28)(21,27)(22,26);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,14,18,22)(15,20,23,27)(16,21,25,28)(17,26,24,19);(2,4)(5,7,9,8)(6,11,12,10)(13,15)(14,20)(16,24)(17,25)(18,23)(19,21)(22,27)(26,28);(2,4)(5,8,9,7)(6,11,12,10)(13,16,18,25)(14,21,22,28)(15,24,23,17)(19,27,26,20);(15,23)(16,25)(20,27)(21,28)Polycyclic presentation:
2^9|b^2=g,c^2=gi,d^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,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, 6360119);