Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator | C2×Q8 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C2×#3021 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
2028 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6,10,12,11,16,8,15)(7,14,18,20,9,13,17,19);(1,2,3,4)(6,12)(8,10)(13,19)(14,20)(15,16)(17,18);(6,13)(7,9)(8,10)(12,19)(14,16)(15,20);(5,7,11,9)(6,14,16,13)(8,17,10,18)(12,20,15,19)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fhi,h^a=hi,e^b=ei,f^b=fhi,g^b=gi,d^c=di,e^c=eh,f^c=fi,g^c=gh,f^e=fi,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72215601, #72233839, #72234081, … |
| maximal normal subgroup of | #72215601, #72233839, #72234081, … |
| maximal quotient of | #72213489, #72231177, #72231376, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401818);