Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535655 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2102 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)(13,14)(15,24)(16,21)(17,22)(18,27)(19,23)(20,26)(25,28);(5,6)(7,10)(8,9)(11,12)(13,15,18,26,19,25,17,21)(14,20,22,24,23,16,27,28);(2,4)(6,9)(10,12)(15,25)(16,20)(21,26)(24,28);(5,7)(6,10)(8,11)(9,12)(13,16)(14,21)(15,22)(17,24)(18,28)(19,20)(23,26)(25,27);(7,11)(10,12)(13,17,19,18)(14,22,23,27)(15,21,25,26)(16,28,20,24)Polycyclic presentation:
2^9|a^2=g,b^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,d^c=di,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6452394);