Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470161 |
| 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×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3186 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)(18,21)(19,22)(20,27)(26,28);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,26)(14,20,22,16)(17,25,18,27)(21,28,23,24);(6,9)(10,12)(13,16,19,20)(14,15,22,26)(17,24,18,28)(21,25,23,27);(6,9)(10,12)(13,17,19,18)(14,21,22,23)(15,25,26,27)(16,24,20,28);(5,7,8,11)(6,10,9,12)(14,22)(15,26)(21,23)(25,27)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,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, 6398295);