Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12753 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×D4 | C2×C42 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C2×#534 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
1221 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6)(7,13,15,11)(8,16,18,10)(9,12)(14,19)(17,20);(1,2,3,4)(6,10)(11,19)(12,16)(13,20);(5,7)(6,11)(8,17)(9,15)(10,19)(12,13)(14,18)(16,20)Polycyclic presentation:
2^9|a^2=f,b^2=g,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,e^b=eh,f^b=fh,d^c=dhi,f^c=fi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72233086, #72233100, #72233141, … |
| maximal normal subgroup of | #72233086, #72233100, #72233141, … |
| maximal quotient of | #72233083, #72233098, #72233639, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53810);