Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996607 |
| 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 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2068 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,7,9,10,11,12)(13,14,17,20)(15,18,21,25)(16,23,24,19)(22,28,27,26);(5,6)(7,11)(8,12)(9,10)(15,21)(16,24)(18,25)(19,23);(1,2,3,4)(6,10)(7,12)(13,15,17,21)(14,18,20,25)(16,22,24,27)(19,26,23,28);(5,7,11,6,9,12,8,10)(15,21)(18,25)(22,27)(26,28);(6,10)(7,12)(13,16,17,24)(14,19,20,23)(15,22,21,27)(18,26,25,28)Polycyclic presentation:
2^9|a^2=fhi,c^2=gh,d^2=fi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7104060);