Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #400057 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2631 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1441 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(6,12)(7,16)(8,13)(9,11)(14,15)(19,20);(1,2)(3,4)(5,6)(7,8)(9,14)(10,15)(11,12)(13,17)(16,20)(18,19);(2,4)(5,7)(6,13)(8,12)(9,17)(10,16)(11,18)(14,20)(15,19);(5,8,11,19,10,20,9,13)(6,7,12,18,15,16,14,17)Polycyclic presentation:
2^9|d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224753, #72224756, #72224757 |
| maximal normal subgroup of | #72224753, #72224756, #72224757 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 307253);