Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454002 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3003 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2353 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(5,6,10,14)(7,15,18,20)(8,11,9,13)(12,17,19,16);(6,11)(7,16)(8,9)(13,14)(15,20)(17,18);(2,4)(5,7)(6,12)(8,17)(9,16)(10,18)(11,15)(13,20)(14,19);(6,13)(7,17)(8,9)(11,14)(12,19)(16,18)Polycyclic presentation:
2^9|a^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72215520, #72215912, #72215936, … |
| maximal normal subgroup of | #72215520, #72215912, #72215936, … |
| maximal quotient of | #72236779, #72237077, #72237191, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361890);