Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470003 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3449 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)(15,22)(16,21)(17,24)(18,23)(19,25)(20,26)(27,28);(5,6)(7,10)(8,9)(11,12)(13,15,19,27)(14,20,25,21)(16,17,26,18)(22,24,28,23);(2,4)(6,9)(10,12)(13,16)(14,21)(15,18)(17,27)(19,26)(20,25)(22,23)(24,28);(6,9)(10,12)(13,15)(14,22)(16,17)(18,26)(19,27)(20,23)(21,24)(25,28);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,23,25,24)(15,26,27,16)(20,22,21,28)Polycyclic presentation:
2^9|b^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398125);