Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432861 |
| 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 | |
3455 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)(16,19)(17,21)(18,26)(20,24)(23,27)(25,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,23)(17,24)(18,27)(19,25)(21,28)(22,26);(2,4)(6,9)(10,12)(13,14,19,16)(15,23,25,20)(17,26,18,21)(22,24,28,27);(5,7,8,11)(6,10,9,12)(13,16,19,14)(15,23,25,20)(17,21,18,26)(22,24,28,27);(13,17,19,18)(14,21,16,26)(15,24,25,27)(20,28,23,22)Polycyclic presentation:
2^9|c^2=i,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,f^b=fi,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, 6342241);