Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×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 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3206 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14,19,15)(16,18,23,20)(17,21,25,22)(24,26,28,27);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,15,18,23,19,14,20,16)(17,21,26,24,25,22,27,28);(6,10)(7,11)(13,16,18,15,19,23,20,14)(17,24,26,22,25,28,27,21);(7,11)(9,12)(13,17)(14,21)(15,22)(16,24)(18,26)(19,25)(20,27)(23,28);(17,25)(21,22)(24,28)(26,27)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=ghi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6366971);