Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C212×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8798 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(14,17)(18,20);(13,14,16,17)(15,18,19,20);(6,9)(10,12)(13,15,16,19)(14,18,17,20);(7,11)(10,12);(5,7)(6,10)(8,11)(9,12);(5,6)(7,10)(8,9)(11,12)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=g,b^a=bg,c^a=ch,f^c=fh,e^d=eh170 conjugacy classes — view the full character table.
| maximal subgroup of | #72233756, #72233760, #72234460 |
| maximal normal subgroup of | #72233756, #72233760, #72234460 |
| maximal quotient of | #72263747 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532970);