Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460861 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2002 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1224 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)(7,13)(8,16)(10,17)(18,20);(6,10,12,17)(11,18,14,20);(1,2,3,4)(5,6)(7,14,13,11)(8,17)(9,12)(10,16)(15,18,19,20);(5,7,8,15,9,13,16,19)(6,11,17,20,12,14,10,18)Polycyclic presentation:
2^9|b^2=e,c^2=f,d^2=eg,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^c=de,e^c=ei,g^c=gi,h^c=hi,f^d=fi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72319417, #72319453, #72319478, … |
| maximal normal subgroup of | #72319417, #72319453, #72319478, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 376964);