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×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2281 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6)(7,10)(8,11)(9,12)(13,14,17,16,18,20,15,19);(5,6)(7,10)(8,11)(9,12)(14,19)(15,17)(16,20);(1,2,3,4)(5,7,10,6)(8,11,9,12)(13,15,18,17)(14,16,20,19);(5,8,10,9)(6,11,7,12)(13,16,15,14,18,19,17,20);(5,9,10,8)(6,11,7,12)(13,15,18,17)(14,16,20,19)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=fh,e^2=h,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.
| maximal subgroup of | #72282978, #72283199, #72283615, … |
| maximal normal subgroup of | #72282978, #72283199, #72283615, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7106266);