Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417674 |
| 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 | |
2415 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,15)(8,12)(9,13)(10,14)(11,16)(17,20)(18,19);(1,2,3,4)(5,7,10,18)(6,11,14,20)(8,17,9,16)(12,15,13,19);(6,12)(7,16)(8,9)(13,14)(15,19)(17,18);(2,4)(6,12)(7,17)(8,9)(13,14)(15,19)(16,18)Polycyclic presentation:
2^9|b^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,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72215520, #72215972, #72216047, … |
| maximal normal subgroup of | #72215520, #72215972, #72216047, … |
| maximal quotient of | #72238739, #72238747, C2×#421068, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 328264);