Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994043 |
| 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 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2368 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)(7,11)(10,12)(13,14,18,21)(15,22,24,27)(16,20,17,26)(19,25,28,23);(6,9)(7,11)(13,15,16,25,18,24,17,23)(14,19,26,27,21,28,20,22);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,17)(14,20,21,26)(15,23,24,25)(19,27,28,22);(6,9)(10,12)(15,23)(16,17)(19,22)(20,26)(24,25)(27,28);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7050798);