Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455822 |
| 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 | |
1273 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)(6,10)(7,14)(9,11)(12,19)(15,16)(17,20);(5,6,11,18,8,10,9,13)(7,12,14,20,15,17,16,19);(2,4)(9,11)(12,17)(13,18)(14,16);(5,7)(6,12)(8,15)(9,16)(10,17)(11,14)(13,19)(18,20)Polycyclic presentation:
2^9|b^2=egh,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,g^b=gi,d^c=de,h^c=hi,f^d=fi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72319859, #72320018, #72320075, … |
| maximal normal subgroup of | #72319859, #72320018, #72320075, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378630);