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 | |
4231 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,24)(16,26)(17,22)(18,28)(19,23)(20,25)(21,27);(2,4)(5,7)(6,9)(8,11)(10,12)(13,15,19,16)(14,20,23,21)(17,25,18,27)(22,26,28,24);(7,11)(9,12)(13,16,19,15)(14,21,23,20)(17,27,18,25)(22,24,28,26);(6,10)(9,12)(13,14)(15,20)(16,21)(17,28)(18,22)(19,23)(24,25)(26,27);(13,17,19,18)(14,22,23,28)(15,25,16,27)(20,26,21,24)Polycyclic presentation:
2^9|b^2=i,c^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340139);